{"id":8946,"date":"2017-06-05T21:56:09","date_gmt":"2017-06-05T21:56:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&p=8946"},"modified":"2020-07-02T05:42:55","modified_gmt":"2020-07-02T05:42:55","slug":"problem-set-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/chapter\/problem-set-polynomials\/","title":{"raw":"Problem Set: Polynomials","rendered":"Problem Set: Polynomials"},"content":{"raw":"

Practice Makes Perfect<\/h2>\r\nIdentify Polynomials, Monomials, Binomials and Trinomials<\/strong>\r\nIn the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial.\r\n\r\n[latex]5x+2[\/latex]\r\n\r\nbinomial\r\n\r\n[latex]{z}^{2}-5z - 6[\/latex]\r\n\r\n[latex]{a}^{2}+9a+18[\/latex]\r\n\r\ntrinomial\r\n\r\n[latex]-12{p}^{4}[\/latex]\r\n\r\n[latex]{y}^{3}-8{y}^{2}+2y - 16[\/latex]\r\n\r\npolynomial\r\n\r\n[latex]10 - 9x[\/latex]\r\n\r\n[latex]23{y}^{2}[\/latex]\r\n\r\nmonomial\r\n\r\n[latex]{m}^{4}+4{m}^{3}+6{m}^{2}+4m+1[\/latex]\r\n\r\nDetermine the Degree of Polynomials<\/strong>\r\nIn the following exercises, determine the degree of each polynomial.\r\n\r\n[latex]8{a}^{5}-2{a}^{3}+1[\/latex]\r\n\r\n5\r\n\r\n[latex]5{c}^{3}+11{c}^{2}-c - 8[\/latex]\r\n\r\n[latex]3x - 12[\/latex]\r\n\r\n1\r\n\r\n[latex]4y+17[\/latex]\r\n\r\n[latex]-13[\/latex]\r\n\r\n0\r\n\r\n[latex]-22[\/latex]\r\n\r\nAdd and Subtract Monomials<\/strong>\r\nIn the following exercises, add or subtract the monomials.\r\n\r\n[latex]{\\text{6x}}^{2}+9{x}^{2}[\/latex]\r\n\r\n15x<\/em>2<\/sup>\r\n\r\n[latex]{\\text{4y}}^{3}+6{y}^{3}[\/latex]\r\n\r\n[latex]-12u+4u[\/latex]\r\n\r\n\u22128u<\/em>\r\n\r\n[latex]-3m+9m[\/latex]\r\n\r\n[latex]5a+7b[\/latex]\r\n\r\n5a<\/em> + 7b<\/em>\r\n\r\n[latex]8y+6z[\/latex]\r\n\r\nAdd: [latex]\\text{}4a,-3b,-8a[\/latex]\r\n\r\n\u22124a<\/em> \u22123b<\/em>\r\n\r\nAdd: [latex]4x,3y,-3x[\/latex]\r\n\r\n[latex]18x - 2x[\/latex]\r\n\r\n16x<\/em>\r\n\r\n[latex]13a - 3a[\/latex]\r\n\r\nSubtract [latex]5{x}^{6}\\text{from}-12{x}^{6}[\/latex]\r\n\r\n\u221217x<\/em>6<\/sup>\r\n\r\nSubtract [latex]2{p}^{4}\\text{from}-7{p}^{4}[\/latex]\r\n\r\nAdd and Subtract Polynomials<\/strong>\r\nIn the following exercises, add or subtract the polynomials.\r\n\r\n[latex]\\left(4{y}^{2}+10y+3\\right)+\\left(8{y}^{2}-6y+5\\right)[\/latex]\r\n\r\n12y<\/em>2<\/sup> + 4y<\/em> + 8\r\n\r\n[latex]\\left(7{x}^{2}-9x+2\\right)+\\left(6{x}^{2}-4x+3\\right)[\/latex]\r\n\r\n[latex]\\left({x}^{2}+6x+8\\right)+\\left(-4{x}^{2}+11x - 9\\right)[\/latex]\r\n\r\n\u22123x<\/em>2<\/sup> + 17x<\/em> \u2212 1\r\n\r\n[latex]\\left({y}^{2}+9y+4\\right)+\\left(-2{y}^{2}-5y - 1\\right)[\/latex]\r\n\r\n[latex]\\left(3{a}^{2}+7\\right)+\\left({a}^{2}-7a - 18\\right)[\/latex]\r\n\r\n4a<\/em>2<\/sup> \u2212 7a<\/em> \u2212 11\r\n\r\n[latex]\\left({p}^{2}-5p - 11\\right)+\\left(3{p}^{2}+9\\right)[\/latex]\r\n\r\n[latex]\\left(6{m}^{2}-9m - 3\\right)-\\left(2{m}^{2}+m - 5\\right)[\/latex]\r\n\r\n4m<\/em>2<\/sup> \u2212 10m<\/em> + 2\r\n\r\n[latex]\\left(3{n}^{2}-4n+1\\right)-\\left(4{n}^{2}-n - 2\\right)[\/latex]\r\n\r\n[latex]\\left({z}^{2}+8z+9\\right)-\\left({z}^{2}-3z+1\\right)[\/latex]\r\n\r\n11z<\/em> + 8\r\n\r\n[latex]\\left({z}^{2}-7z+5\\right)-\\left({z}^{2}-8z+6\\right)[\/latex]\r\n\r\n[latex]\\left(12{s}^{2}-15s\\right)-\\left(s - 9\\right)[\/latex]\r\n\r\n12s<\/em>2<\/sup> \u2212 16s<\/em> + 9\r\n\r\n[latex]\\left(10{r}^{2}-20r\\right)-\\left(r - 8\\right)[\/latex]\r\n\r\nFind the sum of [latex]\\left(2{p}^{3}-8\\right)[\/latex] and [latex]\\left({p}^{2}+9p+18\\right)[\/latex]\r\n\r\n2p<\/em>3<\/sup> + p<\/em>2<\/sup> + 9p<\/em> + 10\r\n\r\nFind the sum of [latex]\\left({q}^{2}+4q+13\\right)[\/latex] and [latex]\\left(7{q}^{3}-3\\right)[\/latex]\r\n\r\nSubtract [latex]\\left(7{x}^{2}-4x+2\\right)[\/latex] from [latex]\\left(8{x}^{2}-x+6\\right)[\/latex]\r\n\r\nx<\/em>2<\/sup> + 3x<\/em> + 4\r\n\r\nSubtract [latex]\\left(5{x}^{2}-x+12\\right)[\/latex] from [latex]\\left(9{x}^{2}-6x - 20\\right)[\/latex]\r\n\r\nFind the difference of [latex]\\left({w}^{2}+w - 42\\right)[\/latex] and [latex]\\left({w}^{2}-10w+24\\right)[\/latex]\r\n\r\n11w<\/em> \u2212 66\r\n\r\nFind the difference of [latex]\\left({z}^{2}-3z - 18\\right)[\/latex] and [latex]\\left({z}^{2}+5z - 20\\right)[\/latex]\r\n\r\nEvaluate a Polynomial for a Given Value<\/strong>\r\nIn the following exercises, evaluate each polynomial for the given value.\r\n\r\n[latex]\\text{Evaluate}8{y}^{2}-3y+2[\/latex]\r\n\r\n\u24d0 [latex]y=5[\/latex]\r\n\u24d1 [latex]y=-2[\/latex]\r\n\u24d2 [latex]y=0[\/latex]\r\n\r\n\u24d0 187\r\n\u24d1 40\r\n\u24d2 2\r\n\r\n[latex]\\text{Evaluate}5{y}^{2}-y - 7\\text{when:}[\/latex]\r\n\r\n\u24d0 [latex]y=-4[\/latex]\r\n\u24d1 [latex]y=1[\/latex]\r\n\u24d2 [latex]y=0[\/latex]\r\n\r\n[latex]\\text{Evaluate}4 - 36x\\text{when:}[\/latex]\r\n\r\n\u24d0 [latex]x=3[\/latex]\r\n\u24d1 [latex]x=0[\/latex]\r\n\u24d2 [latex]x=-1[\/latex]\r\n\r\n\u24d0 \u2212104\r\n\u24d1 4\r\n\u24d2 40\r\n\r\n[latex]\\text{Evaluate}16 - 36{x}^{2}\\text{when:}[\/latex]\r\n\r\n\u24d0 [latex]x=-1[\/latex]\r\n\u24d1 [latex]x=0[\/latex]\r\n\u24d2 [latex]x=2[\/latex]\r\n\r\nA window washer drops a squeegee from a platform [latex]275[\/latex] feet high. The polynomial [latex]-16{t}^{2}+275[\/latex] gives the height of the squeegee [latex]t[\/latex] seconds after it was dropped. Find the height after [latex]t=4[\/latex] seconds.\r\n\r\n19 feet\r\n\r\nA manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p<\/em> dollars each is given by the polynomial [latex]-5{p}^{2}+350p[\/latex]. Find the revenue received when [latex]p=50[\/latex] dollars.\r\n

Everyday Math<\/h2>\r\nFuel Efficiency<\/strong> The fuel efficiency (in miles per gallon) of a bus going at a speed of [latex]x[\/latex] miles per hour is given by the polynomial [latex]-\\Large\\frac{1}{160}\\normalsize{x}^{2}+\\Large\\frac{1}{2}\\normalsize x[\/latex]. Find the fuel efficiency when [latex]x=40\\text{mph.}[\/latex]\r\n\r\n10 mpg\r\n\r\nStopping Distance<\/strong> The number of feet it takes for a car traveling at [latex]x[\/latex] miles per hour to stop on dry, level concrete is given by the polynomial [latex]0.06{x}^{2}+1.1x[\/latex]. Find the stopping distance when [latex]x=60\\text{mph.}[\/latex]\r\n

Writing Exercises<\/h2>\r\nUsing your own words, explain the difference between a monomial, a binomial, and a trinomial.\r\n\r\nAnswers will vary.\r\n\r\nEloise thinks the sum [latex]5{x}^{2}+3{x}^{4}[\/latex] is [latex]8{x}^{6}[\/latex]. What is wrong with her reasoning?\r\n

Self Check<\/h2>\r\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\r\n\r\n\".\"\r\n\u24d1 If most of your checks were:\r\n\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.\r\n\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?\r\n\u2026no\u2014I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.\r\n

<\/h2>\r\n

Practice Makes Perfect<\/h2>\r\nSimplify Expressions with Exponents<\/strong>\r\nIn the following exercises, simplify each expression with exponents.\r\n\r\n[latex]{4}^{5}[\/latex]\r\n\r\n1,024\r\n\r\n[latex]{10}^{3}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{4}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{3}{5}\\normalsize\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(0.2\\right)}^{3}[\/latex]\r\n\r\n0.008\r\n\r\n[latex]{\\left(0.4\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(-5\\right)}^{4}[\/latex]\r\n\r\n625\r\n\r\n[latex]{\\left(-3\\right)}^{5}[\/latex]\r\n\r\n[latex]{-5}^{4}[\/latex]\r\n\r\n\u2212625\r\n\r\n[latex]{-3}^{5}[\/latex]\r\n\r\n[latex]{-10}^{4}[\/latex]\r\n\r\n\u221210,000\r\n\r\n[latex]{-2}^{6}[\/latex]\r\n\r\n[latex]{\\left(-\\Large\\frac{2}{3}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]-\\Large\\frac{8}{27}[\/latex]\r\n\r\n[latex]{\\left(-\\Large\\frac{1}{4}\\normalsize\\right)}^{4}[\/latex]\r\n\r\n[latex]-{0.5}^{2}[\/latex]\r\n\r\n\u2212.25\r\n\r\n[latex]-{0.1}^{4}[\/latex]\r\n\r\nSimplify Expressions Using the Product Property of Exponents<\/strong>\r\nIn the following exercises, simplify each expression using the Product Property of Exponents.\r\n\r\n[latex]{x}^{3}\\cdot {x}^{6}[\/latex]\r\n\r\nx<\/em>9<\/sup>\r\n\r\n[latex]{m}^{4}\\cdot {m}^{2}[\/latex]\r\n\r\n[latex]a\\cdot {a}^{4}[\/latex]\r\n\r\na<\/em>5<\/sup>\r\n\r\n[latex]{y}^{12}\\cdot y[\/latex]\r\n\r\n[latex]{3}^{5}\\cdot {3}^{9}[\/latex]\r\n\r\n314<\/sup>\r\n\r\n[latex]{5}^{10}\\cdot {5}^{6}[\/latex]\r\n\r\n[latex]z\\cdot {z}^{2}\\cdot {z}^{3}[\/latex]\r\n\r\nz<\/em>6<\/sup>\r\n\r\n[latex]a\\cdot {a}^{3}\\cdot {a}^{5}[\/latex]\r\n\r\n[latex]{x}^{a}\\cdot {x}^{2}[\/latex]\r\n\r\nx<\/em>a<\/em>+2<\/sup>\r\n\r\n[latex]{y}^{p}\\cdot {y}^{3}[\/latex]\r\n\r\n[latex]{y}^{a}\\cdot {y}^{b}[\/latex]\r\n\r\ny<\/em>a<\/em>+b<\/em><\/sup>\r\n\r\n[latex]{x}^{p}\\cdot {x}^{q}[\/latex]\r\n\r\nSimplify Expressions Using the Power Property of Exponents<\/strong>\r\nIn the following exercises, simplify each expression using the Power Property of Exponents.<\/em>\r\n\r\n[latex]{\\left({u}^{4}\\right)}^{2}[\/latex]\r\n\r\nu<\/em>8<\/sup>\r\n\r\n[latex]{\\left({x}^{2}\\right)}^{7}[\/latex]\r\n\r\n[latex]{\\left({y}^{5}\\right)}^{4}[\/latex]\r\n\r\ny<\/em>20<\/sup>\r\n\r\n[latex]{\\left({a}^{3}\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left({10}^{2}\\right)}^{6}[\/latex]\r\n\r\n1012<\/sup>\r\n\r\n[latex]{\\left({2}^{8}\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left({x}^{15}\\right)}^{6}[\/latex]\r\n\r\nx<\/em>90<\/sup>\r\n\r\n[latex]{\\left({y}^{12}\\right)}^{8}[\/latex]\r\n\r\n[latex]{\\left({x}^{2}\\right)}^{y}[\/latex]\r\n\r\nx<\/em>2y<\/em><\/sup>\r\n\r\n[latex]{\\left({y}^{3}\\right)}^{x}[\/latex]\r\n\r\n[latex]{\\left({5}^{x}\\right)}^{y}[\/latex]\r\n\r\n5x<\/em>y<\/em><\/sup>\r\n\r\n[latex]{\\left({7}^{a}\\right)}^{b}[\/latex]\r\n\r\nSimplify Expressions Using the Product to a Power Property<\/strong>\r\nIn the following exercises, simplify each expression using the Product to a Power Property.\r\n\r\n[latex]{\\left(5a\\right)}^{2}[\/latex]\r\n\r\n25a<\/em>2<\/sup>\r\n\r\n[latex]{\\left(7x\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(-6m\\right)}^{3}[\/latex]\r\n\r\n\u2212216m<\/em>3<\/sup>\r\n\r\n[latex]{\\left(-9n\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(4rs\\right)}^{2}[\/latex]\r\n\r\n16r<\/em>2<\/sup>s<\/em>2<\/sup>\r\n\r\n[latex]{\\left(5ab\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(4xyz\\right)}^{4}[\/latex]\r\n\r\n256x<\/em>4<\/sup>y<\/em>4<\/sup>z<\/em>4<\/sup>\r\n\r\n[latex]{\\left(-5abc\\right)}^{3}[\/latex]\r\n\r\nSimplify Expressions by Applying Several Properties<\/strong>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]{\\left({x}^{2}\\right)}^{4}\\cdot {\\left({x}^{3}\\right)}^{2}[\/latex]\r\n\r\nx<\/em>14<\/sup>\r\n\r\n[latex]{\\left({y}^{4}\\right)}^{3}\\cdot {\\left({y}^{5}\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left({a}^{2}\\right)}^{6}\\cdot {\\left({a}^{3}\\right)}^{8}[\/latex]\r\n\r\na<\/em>36<\/sup>\r\n\r\n[latex]{\\left({b}^{7}\\right)}^{5}\\cdot {\\left({b}^{2}\\right)}^{6}[\/latex]\r\n\r\n[latex]{\\left(3x\\right)}^{2}\\left(5x\\right)[\/latex]\r\n\r\n45x<\/em>3<\/sup>\r\n\r\n[latex]{\\left(2y\\right)}^{3}\\left(6y\\right)[\/latex]\r\n\r\n[latex]{\\left(5a\\right)}^{2}{\\left(2a\\right)}^{3}[\/latex]\r\n\r\n200a<\/em>5<\/sup>\r\n\r\n[latex]{\\left(4b\\right)}^{2}{\\left(3b\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(2{m}^{6}\\right)}^{3}[\/latex]\r\n\r\n8m<\/em>18<\/sup>\r\n\r\n[latex]{\\left(3{y}^{2}\\right)}^{4}[\/latex]\r\n\r\n[latex]{\\left(10{x}^{2}y\\right)}^{3}[\/latex]\r\n\r\n1,000x<\/em>6<\/sup>y<\/em>3<\/sup>\r\n\r\n[latex]{\\left(2m{n}^{4}\\right)}^{5}[\/latex]\r\n\r\n[latex]{\\left(-2{a}^{3}{b}^{2}\\right)}^{4}[\/latex]\r\n\r\n16a<\/em>12<\/sup>b<\/em>8<\/sup>\r\n\r\n[latex]{\\left(-10{u}^{2}{v}^{4}\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{2}{3}\\normalsize{x}^{2}y\\right)}^{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{8}{27}\\normalsize{x}^{6}{y}^{3}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{7}{9}\\normalsize p{q}^{4}\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(8{a}^{3}\\right)}^{2}{\\left(2a\\right)}^{4}[\/latex]\r\n\r\n1,024a<\/em>10<\/sup>\r\n\r\n[latex]{\\left(5{r}^{2}\\right)}^{3}{\\left(3r\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(10{p}^{4}\\right)}^{3}{\\left(5{p}^{6}\\right)}^{2}[\/latex]\r\n\r\n25,000p<\/em>24<\/sup>\r\n\r\n[latex]{\\left(4{x}^{3}\\right)}^{3}{\\left(2{x}^{5}\\right)}^{4}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{1}{2}\\normalsize{x}^{2}{y}^{3}\\right)}^{4}{\\left(4{x}^{5}{y}^{3}\\right)}^{2}[\/latex]\r\n\r\nx<\/em>18<\/sup>y<\/em>18<\/sup>\r\n\r\n[latex]{\\left(\\Large\\frac{1}{3}\\normalsize{m}^{3}{n}^{2}\\right)}^{4}{\\left(9{m}^{8}{n}^{3}\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(3{m}^{2}n\\right)}^{2}{\\left(2m{n}^{5}\\right)}^{4}[\/latex]\r\n\r\n144m<\/em>8<\/sup>n<\/em>22<\/sup>\r\n\r\n[latex]{\\left(2p{q}^{4}\\right)}^{3}{\\left(5{p}^{6}q\\right)}^{2}[\/latex]\r\n\r\nMultiply Monomials<\/strong>\r\nIn the following exercises, multiply the following monomials.\r\n\r\n[latex]\\left(12{x}^{2}\\right)\\left(-5{x}^{4}\\right)[\/latex]\r\n\r\n\u221260x<\/em>6<\/sup>\r\n\r\n[latex]\\left(-10{y}^{3}\\right)\\left(7{y}^{2}\\right)[\/latex]\r\n\r\n[latex]\\left(-8{u}^{6}\\right)\\left(-9u\\right)[\/latex]\r\n\r\n72u<\/em>7<\/sup>\r\n\r\n[latex]\\left(-6{c}^{4}\\right)\\left(-12c\\right)[\/latex]\r\n\r\n[latex]\\left(\\Large\\frac{1}{5}\\normalsize{r}^{8}\\right)\\left(20{r}^{3}\\right)[\/latex]\r\n\r\n4r<\/em>11<\/sup>\r\n\r\n[latex]\\left(\\Large\\frac{1}{4}\\normalsize{a}^{5}\\right)\\left(36{a}^{2}\\right)[\/latex]\r\n\r\n[latex]\\left(4{a}^{3}b\\right)\\left(9{a}^{2}{b}^{6}\\right)[\/latex]\r\n\r\n36a<\/em>5<\/sup>b<\/em>7<\/sup>\r\n\r\n[latex]\\left(6{m}^{4}{n}^{3}\\right)\\left(7m{n}^{5}\\right)[\/latex]\r\n\r\n[latex]\\left(\\Large\\frac{4}{7}\\normalsize x{y}^{2}\\right)\\left(14x{y}^{3}\\right)[\/latex]\r\n\r\n8x<\/em>2<\/sup>y<\/em>5<\/sup>\r\n\r\n[latex]\\left(\\Large\\frac{5}{8}\\normalsize{u}^{3}v\\right)\\left(24{u}^{5}v\\right)[\/latex]\r\n\r\n[latex]\\left(\\Large\\frac{2}{3}\\normalsize{x}^{2}y\\right)\\left(\\Large\\frac{3}{4}\\normalsize x{y}^{2}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{2}\\normalsize{x}^{3}{y}^{3}[\/latex]\r\n\r\n[latex]\\left(\\Large\\frac{3}{5}\\normalsize{m}^{3}{n}^{2}\\right)\\left(\\Large\\frac{5}{9}\\normalsize{m}^{2}{n}^{3}\\right)[\/latex]\r\n

Everyday Math<\/h2>\r\nEmail<\/strong> Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is [latex]{6}^{2}[\/latex], on the third round is [latex]{6}^{3}[\/latex], as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Round<\/th>\r\nNumber of people<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]{6}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]{6}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\dots [\/latex]<\/td>\r\n[latex]\\dots [\/latex]<\/td>\r\n<\/tr>\r\n
[latex]8[\/latex]<\/td>\r\n[latex]?[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n1,679,616\r\n\r\nSalary<\/strong> Raul\u2019s boss gives him a [latex]\\text{5%}[\/latex] raise every year on his birthday. This means that each year, Raul\u2019s salary is [latex]1.05[\/latex] times his last year\u2019s salary. If his original salary was [latex]{$40,000}[\/latex] , his salary after [latex]1[\/latex] year was [latex]{$40,000}\\left(1.05\\right)[\/latex], after [latex]2[\/latex] years was [latex]{$40,000}{\\left(1.05\\right)}^{2}[\/latex], after [latex]3[\/latex] years was [latex]{$40,000}{\\left(1.05\\right)}^{3}[\/latex], as shown in the table below. What will Raul\u2019s salary be after [latex]10[\/latex] years? Simplify the expression, to show Raul\u2019s salary in dollars.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Year<\/th>\r\nSalary<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]{$40,000}\\left(1.05\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]{$40,000}{\\left(1.05\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]{$40,000}{\\left(1.05\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\dots [\/latex]<\/td>\r\n[latex]\\dots [\/latex]<\/td>\r\n<\/tr>\r\n
[latex]10[\/latex]<\/td>\r\n[latex]?[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Writing Exercises<\/h2>\r\nUse the Product Property for Exponents to explain why [latex]x\\cdot x={x}^{2}[\/latex].\r\n\r\nAnswers will vary.\r\n\r\nExplain why [latex]{-5}^{3}={\\left(-5\\right)}^{3}[\/latex] but [latex]{-5}^{4}\\ne {\\left(-5\\right)}^{4}[\/latex].\r\n\r\nJorge thinks [latex]{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}[\/latex] is [latex]1[\/latex]. What is wrong with his reasoning?\r\n\r\nAnswers will vary.\r\n\r\nExplain why [latex]{x}^{3}\\cdot {x}^{5}[\/latex] is [latex]{x}^{8}[\/latex], and not [latex]{x}^{15}[\/latex].\r\n

Self Check<\/h2>\r\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\r\n\r\n\".\"\r\n\u24d1 After reviewing this checklist, what will you do to become confident for all objectives?\r\n\r\n \r\n

Practice Makes Perfect<\/h2>\r\nMultiply a Polynomial by a Monomial<\/strong>\r\nIn the following exercises, multiply.\r\n\r\n[latex]4\\left(x+10\\right)[\/latex]\r\n\r\n4x<\/em> + 40\r\n\r\n[latex]6\\left(y+8\\right)[\/latex]\r\n\r\n[latex]15\\left(r - 24\\right)[\/latex]\r\n\r\n15r<\/em> \u2212 360\r\n\r\n[latex]12\\left(v - 30\\right)[\/latex]\r\n\r\n[latex]-3\\left(m+11\\right)[\/latex]\r\n\r\n\u22123m<\/em> \u2212 33\r\n\r\n[latex]-4\\left(p+15\\right)[\/latex]\r\n\r\n[latex]-8\\left(z - 5\\right)[\/latex]\r\n\r\n\u22128z<\/em> + 40\r\n\r\n[latex]-3\\left(x - 9\\right)[\/latex]\r\n\r\n[latex]u\\left(u+5\\right)[\/latex]\r\n\r\nu<\/em>2<\/sup> + 5u<\/em>\r\n\r\n[latex]q\\left(q+7\\right)[\/latex]\r\n\r\n[latex]n\\left({n}^{2}-3n\\right)[\/latex]\r\n\r\nn<\/em>3<\/sup> \u2212 3n<\/em>2<\/sup>\r\n\r\n[latex]s\\left({s}^{2}-6s\\right)[\/latex]\r\n\r\n[latex]12x\\left(x - 10\\right)[\/latex]\r\n\r\n12x<\/em>2<\/sup> \u2212 120x<\/em>\r\n\r\n[latex]9m\\left(m - 11\\right)[\/latex]\r\n\r\n[latex]-9a\\left(3a+5\\right)[\/latex]\r\n\r\n\u221227a<\/em>2<\/sup> \u2212 45a<\/em>\r\n\r\n[latex]-4p\\left(2p+7\\right)[\/latex]\r\n\r\n[latex]6x\\left(4x+y\\right)[\/latex]\r\n\r\n24x<\/em>2<\/sup> + 6xy<\/em>\r\n\r\n[latex]5a\\left(9a+b\\right)[\/latex]\r\n\r\n[latex]5p\\left(11p - 5q\\right)[\/latex]\r\n\r\n55p<\/em>2<\/sup> \u2212 25pq<\/em>\r\n\r\n[latex]12u\\left(3u - 4v\\right)[\/latex]\r\n\r\n[latex]3\\left({v}^{2}+10v+25\\right)[\/latex]\r\n\r\n3v<\/em>2<\/sup> + 30v<\/em> + 75\r\n\r\n[latex]6\\left({x}^{2}+8x+16\\right)[\/latex]\r\n\r\n[latex]2n\\left(4{n}^{2}-4n+1\\right)[\/latex]\r\n\r\n8n<\/em>3<\/sup> \u2212 8n<\/em>2<\/sup> + 2n<\/em>\r\n\r\n[latex]3r\\left(2{r}^{2}-6r+2\\right)[\/latex]\r\n\r\n[latex]-8y\\left({y}^{2}+2y - 15\\right)[\/latex]\r\n\r\n\u22128y<\/em>3<\/sup> \u2212 16y<\/em>2<\/sup> + 120y<\/em>\r\n\r\n[latex]-5m\\left({m}^{2}+3m - 18\\right)[\/latex]\r\n\r\n[latex]5{q}^{3}\\left({q}^{2}-2q+6\\right)[\/latex]\r\n\r\n5q<\/em>5<\/sup> \u2212 10q<\/em>4<\/sup> + 30q<\/em>3<\/sup>\r\n\r\n[latex]9{r}^{3}\\left({r}^{2}-3r+5\\right)[\/latex]\r\n\r\n[latex]-4{z}^{2}\\left(3{z}^{2}+12z - 1\\right)[\/latex]\r\n\r\n\u221212z<\/em>4<\/sup> \u2212 48z<\/em>3<\/sup> + 4z<\/em>2<\/sup>\r\n\r\n[latex]-3{x}^{2}\\left(7{x}^{2}+10x - 1\\right)[\/latex]\r\n\r\n[latex]\\left(2y - 9\\right)y[\/latex]\r\n\r\n2y<\/em>2<\/sup> \u2212 9y<\/em>\r\n\r\n[latex]\\left(8b - 1\\right)b[\/latex]\r\n\r\n[latex]\\left(w - 6\\right)\\cdot 8[\/latex]\r\n\r\n8w<\/em> \u2212 48\r\n\r\n[latex]\\left(k - 4\\right)\\cdot 5[\/latex]\r\n\r\nMultiply a Binomial by a Binomial<\/strong>\r\nIn the following exercises, multiply the following binomials using: \u24d0 the Distributive Property \u24d1 the FOIL method \u24d2 the Vertical method\r\n\r\n[latex]\\left(x+4\\right)\\left(x+6\\right)[\/latex]\r\n\r\nx<\/em>2<\/sup> + 10x<\/em> + 24\r\n\r\n[latex]\\left(u+8\\right)\\left(u+2\\right)[\/latex]\r\n\r\n[latex]\\left(n+12\\right)\\left(n - 3\\right)[\/latex]\r\n\r\nn<\/em>2<\/sup> + 9n<\/em> \u2212 36\r\n\r\n[latex]\\left(y+3\\right)\\left(y - 9\\right)[\/latex]\r\n\r\nIn the following exercises, multiply the following binomials. Use any method.\r\n\r\n[latex]\\left(y+8\\right)\\left(y+3\\right)[\/latex]\r\n\r\ny<\/em>2<\/sup> + 11y<\/em> + 24\r\n\r\n[latex]\\left(x+5\\right)\\left(x+9\\right)[\/latex]\r\n\r\n[latex]\\left(a+6\\right)\\left(a+16\\right)[\/latex]\r\n\r\na<\/em>2<\/sup> + 22a<\/em> + 96\r\n\r\n[latex]\\left(q+8\\right)\\left(q+12\\right)[\/latex]\r\n\r\n[latex]\\left(u - 5\\right)\\left(u - 9\\right)[\/latex]\r\n\r\nu<\/em>2<\/sup> \u2212 14u<\/em> + 45\r\n\r\n[latex]\\left(r - 6\\right)\\left(r - 2\\right)[\/latex]\r\n\r\n[latex]\\left(z - 10\\right)\\left(z - 22\\right)[\/latex]\r\n\r\nz<\/em>2<\/sup> \u2212 32z<\/em> + 220\r\n\r\n[latex]\\left(b - 5\\right)\\left(b - 24\\right)[\/latex]\r\n\r\n[latex]\\left(x - 4\\right)\\left(x+7\\right)[\/latex]\r\n\r\nx<\/em>2<\/sup> + 3x<\/em> \u2212 28\r\n\r\n[latex]\\left(s - 3\\right)\\left(s+8\\right)[\/latex]\r\n\r\n[latex]\\left(v+12\\right)\\left(v - 5\\right)[\/latex]\r\n\r\nv<\/em>2<\/sup> + 7v<\/em> \u2212 60\r\n\r\n[latex]\\left(d+15\\right)\\left(d - 4\\right)[\/latex]\r\n\r\n[latex]\\left(6n+5\\right)\\left(n+1\\right)[\/latex]\r\n\r\n6n<\/em>2<\/sup> + 11n<\/em> + 5\r\n\r\n[latex]\\left(7y+1\\right)\\left(y+3\\right)[\/latex]\r\n\r\n[latex]\\left(2m - 9\\right)\\left(10m+1\\right)[\/latex]\r\n\r\n20m<\/em>2<\/sup> \u2212 88m<\/em> \u2212 9\r\n\r\n[latex]\\left(5r - 4\\right)\\left(12r+1\\right)[\/latex]\r\n\r\n[latex]\\left(4c - 1\\right)\\left(4c+1\\right)[\/latex]\r\n\r\n16c<\/em>2<\/sup> \u2212 1\r\n\r\n[latex]\\left(8n - 1\\right)\\left(8n+1\\right)[\/latex]\r\n\r\n[latex]\\left(3u - 8\\right)\\left(5u - 14\\right)[\/latex]\r\n\r\n15u<\/em>2<\/sup> \u2212 82u<\/em> + 112\r\n\r\n[latex]\\left(2q - 5\\right)\\left(7q - 11\\right)[\/latex]\r\n\r\n[latex]\\left(a+b\\right)\\left(2a+3b\\right)[\/latex]\r\n\r\n2a<\/em>2<\/sup> + 5ab<\/em> + 3b<\/em>2<\/sup>\r\n\r\n[latex]\\left(r+s\\right)\\left(3r+2s\\right)[\/latex]\r\n\r\n[latex]\\left(5x-y\\right)\\left(x - 4\\right)[\/latex]\r\n\r\n5x<\/em>2<\/sup> \u2212 20x<\/em> \u2212 xy<\/em> + 4y<\/em>\r\n\r\n[latex]\\left(4z-y\\right)\\left(z - 6\\right)[\/latex]\r\n\r\nMultiply a Trinomial by a Binomial<\/strong>\r\nIn the following exercises, multiply using \u24d0 the Distributive Property and \u24d1 the Vertical Method.\r\n\r\n[latex]\\left(u+4\\right)\\left({u}^{2}+3u+2\\right)[\/latex]\r\n\r\nu<\/em>3<\/sup> + 7u<\/em>2<\/sup> + 14u<\/em> + 8\r\n\r\n[latex]\\left(x+5\\right)\\left({x}^{2}+8x+3\\right)[\/latex]\r\n\r\n[latex]\\left(a+10\\right)\\left(3{a}^{2}+a - 5\\right)[\/latex]\r\n\r\n3a<\/em>3<\/sup> + 31a<\/em>2<\/sup> + 5a<\/em> \u2212 50\r\n\r\n[latex]\\left(n+8\\right)\\left(4{n}^{2}+n - 7\\right)[\/latex]\r\n\r\nIn the following exercises, multiply. Use either method.\r\n\r\n[latex]\\left(y - 6\\right)\\left({y}^{2}-10y+9\\right)[\/latex]\r\n\r\ny<\/em>3<\/sup> \u2212 16y<\/em>2<\/sup> + 69y<\/em> \u2212 54\r\n\r\n[latex]\\left(k - 3\\right)\\left({k}^{2}-8k+7\\right)[\/latex]\r\n\r\n[latex]\\left(2x+1\\right)\\left({x}^{2}-5x - 6\\right)[\/latex]\r\n\r\n2x<\/em>3<\/sup> \u2212 9x<\/em>2<\/sup> \u2212 17x<\/em> \u2212 6\r\n\r\n[latex]\\left(5v+1\\right)\\left({v}^{2}-6v - 10\\right)[\/latex]\r\n

Everyday Math<\/h2>\r\nMental math<\/strong> You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply [latex]13[\/latex] times [latex]15[\/latex]. Think of [latex]13[\/latex] as [latex]10+3[\/latex] and [latex]15[\/latex] as [latex]10+5[\/latex].\r\n
    \r\n \t
  1. \u24d0 Multiply [latex]\\left(10+3\\right)\\left(10+5\\right)[\/latex] by the FOIL method.<\/li>\r\n \t
  2. \u24d1 Multiply [latex]13\\cdot 15[\/latex] without using a calculator.<\/li>\r\n \t
  3. \u24d2 Which way is easier for you? Why?<\/li>\r\n<\/ol>\r\n
      \r\n \t
    1. \u24d0 195<\/li>\r\n \t
    2. \u24d1 195<\/li>\r\n \t
    3. \u24d0 Answers will vary.<\/li>\r\n<\/ol>\r\nMental math<\/strong> You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply [latex]18[\/latex] times [latex]17[\/latex]. Think of [latex]18[\/latex] as [latex]20 - 2[\/latex] and [latex]17[\/latex] as [latex]20 - 3[\/latex].\r\n
        \r\n \t
      1. \u24d0 Multiply [latex]\\left(20 - 2\\right)\\left(20 - 3\\right)[\/latex] by the FOIL method.<\/li>\r\n \t
      2. \u24d1 Multiply [latex]18\\cdot 17[\/latex] without using a calculator.<\/li>\r\n \t
      3. \u24d2 Which way is easier for you? Why?<\/li>\r\n<\/ol>\r\n

        Writing Exercises<\/h2>\r\nWhich method do you prefer to use when multiplying two binomials\u2014the Distributive Property, the FOIL method, or the Vertical Method? Why?\r\n\r\nAnswers will vary.\r\n\r\nWhich method do you prefer to use when multiplying a trinomial by a binomial\u2014the Distributive Property or the Vertical Method? Why?\r\n

        Self Check<\/h2>\r\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\r\n\r\n\".\"\r\n\u24d1 What does this checklist tell you about your mastery of this section? What steps will you take to improve?\r\n

        <\/h2>\r\n

        Practice Makes Perfect<\/h2>\r\nSimplify Expressions Using the Quotient Property of Exponents<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]\\Large\\frac{{4}^{8}}{{4}^{2}}[\/latex]\r\n\r\n46<\/sup>\r\n\r\n[latex]\\Large\\frac{{3}^{12}}{{3}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{x}^{12}}{{x}^{3}}[\/latex]\r\n\r\nx<\/em>9<\/sup>\r\n\r\n[latex]\\Large\\frac{{u}^{9}}{{u}^{3}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{r}^{5}}{r}[\/latex]\r\n\r\nr<\/em>4<\/sup>\r\n\r\n[latex]\\Large\\frac{{y}^{4}}{y}[\/latex]\r\n\r\n[latex]\\Large\\frac{{y}^{4}}{{y}^{20}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{y}^{16}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{x}^{10}}{{x}^{30}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{10}^{3}}{{10}^{15}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{10}^{12}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{r}^{2}}{{r}^{8}}[\/latex]\r\n\r\n[latex]\\Large\\frac{a}{{a}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{a}^{8}}[\/latex]\r\n\r\n[latex]\\Large\\frac{2}{{2}^{5}}[\/latex]\r\n\r\nSimplify Expressions with Zero Exponents<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{5}^{0}[\/latex]\r\n\r\n1\r\n\r\n[latex]{10}^{0}[\/latex]\r\n\r\n[latex]{a}^{0}[\/latex]\r\n\r\n1\r\n\r\n[latex]{x}^{0}[\/latex]\r\n\r\n[latex]-{7}^{0}[\/latex]\r\n\r\n\u22121\r\n\r\n[latex]-{4}^{0}[\/latex]\r\n\r\n\u24d0 [latex]{\\left(10p\\right)}^{0}[\/latex]\r\n\u24d1 [latex]10{p}^{0}[\/latex]\r\n\r\n\u24d0 1\r\n\u24d1 10\r\n\r\n\u24d0 [latex]{\\left(3a\\right)}^{0}[\/latex]\r\n\u24d1 [latex]3{a}^{0}[\/latex]\r\n\r\n\u24d0 [latex]{\\left(-27{x}^{5}y\\right)}^{0}[\/latex]\r\n\u24d1 [latex]-27{x}^{5}{y}^{0}[\/latex]\r\n\r\n\u24d0 1\r\n\u24d1 \u221227x<\/em>5<\/sup>\r\n\r\n\u24d0 [latex]{\\left(-92{y}^{8}z\\right)}^{0}[\/latex]\r\n\u24d1 [latex]-92{y}^{8}{z}^{0}[\/latex]\r\n\r\n\u24d0 [latex]{15}^{0}[\/latex]\r\n\u24d1 [latex]{15}^{1}[\/latex]\r\n\r\n\u24d0 1\r\n\u24d1 15\r\n\r\n\u24d0 [latex]-{6}^{0}[\/latex]\r\n\u24d1 [latex]-{6}^{1}[\/latex]\r\n\r\n[latex]2\\cdot {x}^{0}+5\\cdot {y}^{0}[\/latex]\r\n\r\n7\r\n\r\n[latex]8\\cdot {m}^{0}-4\\cdot {n}^{0}[\/latex]\r\n\r\nSimplify Expressions Using the Quotient to a Power Property<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{\\left(\\Large\\frac{3}{2}\\normalsize\\right)}^{5}[\/latex]\r\n\r\n[latex]\\Large\\frac{243}{32}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{4}{5}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{m}{6}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{{m}^{3}}{216}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{p}{2}\\normalsize\\right)}^{5}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{x}{y}\\normalsize\\right)}^{10}[\/latex]\r\n\r\n[latex]\\Large\\frac{{x}^{10}}{{y}^{10}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{a}{b}\\normalsize\\right)}^{8}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{a}{3b}\\normalsize\\right)}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{{a}^{2}}{9{b}^{2}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{2x}{y}\\normalsize\\right)}^{4}[\/latex]\r\n\r\nSimplify Expressions by Applying Several Properties<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]\\Large\\frac{{\\left({x}^{2}\\right)}^{4}}{{x}^{5}}[\/latex]\r\n\r\nx<\/em>3<\/sup>\r\n\r\n[latex]\\Large\\frac{{\\left({y}^{4}\\right)}^{3}}{{y}^{7}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({u}^{3}\\right)}^{4}}{{u}^{10}}[\/latex]\r\n\r\nu<\/em>2<\/sup>\r\n\r\n[latex]\\Large\\frac{{\\left({y}^{2}\\right)}^{5}}{{y}^{6}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{y}^{8}}{{\\left({y}^{5}\\right)}^{2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{y}^{2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{p}^{11}}{{\\left({p}^{5}\\right)}^{3}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{r}^{5}}{{r}^{4}\\cdot r}[\/latex]\r\n\r\n1\r\n\r\n[latex]\\Large\\frac{{a}^{3}\\cdot {a}^{4}}{{a}^{7}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{{x}^{2}}{{x}^{8}}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{x}^{18}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{u}{{u}^{10}}\\normalsize\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{{a}^{4}\\cdot {a}^{6}}{{a}^{3}}\\normalsize\\right)}^{2}[\/latex]\r\n\r\na<\/em>14<\/sup>\r\n\r\n[latex]{\\left(\\Large\\frac{{x}^{3}\\cdot {x}^{8}}{{x}^{4}}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({y}^{3}\\right)}^{5}}{{\\left({y}^{4}\\right)}^{3}}[\/latex]\r\n\r\ny<\/em>3<\/sup>\r\n\r\n[latex]\\Large\\frac{{\\left({z}^{6}\\right)}^{2}}{{\\left({z}^{2}\\right)}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({x}^{3}\\right)}^{6}}{{\\left({x}^{4}\\right)}^{7}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{x}^{10}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({x}^{4}\\right)}^{8}}{{\\left({x}^{5}\\right)}^{7}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{2{r}^{3}}{5s}\\normalsize\\right)}^{4}[\/latex]\r\n\r\n[latex]\\Large\\frac{16{r}^{12}}{625{s}^{4}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{3{m}^{2}}{4n}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{3{y}^{2}\\cdot {y}^{5}}{{y}^{15}\\cdot {y}^{8}}\\normalsize\\right)}^{0}[\/latex]\r\n\r\n1\r\n\r\n[latex]{\\left(\\Large\\frac{15{z}^{4}\\cdot {z}^{9}}{0.3{z}^{2}}\\normalsize\\right)}^{0}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({r}^{2}\\right)}^{5}{\\left({r}^{4}\\right)}^{2}}{{\\left({r}^{3}\\right)}^{7}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{r}^{3}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({p}^{4}\\right)}^{2}{\\left({p}^{3}\\right)}^{5}}{{\\left({p}^{2}\\right)}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left(3{x}^{4}\\right)}^{3}{\\left(2{x}^{3}\\right)}^{2}}{{\\left(6{x}^{5}\\right)}^{2}}[\/latex]\r\n\r\n3x<\/em>8<\/sup>\r\n\r\n[latex]\\Large\\frac{{\\left(-2{y}^{3}\\right)}^{4}{\\left(3{y}^{4}\\right)}^{2}}{{\\left(-6{y}^{3}\\right)}^{2}}[\/latex]\r\n\r\nDivide Monomials<\/strong>\r\nIn the following exercises, divide the monomials.\r\n\r\n[latex]48{b}^{8}\\div 6{b}^{2}[\/latex]\r\n\r\n8b<\/em>6<\/sup>\r\n\r\n[latex]42{a}^{14}\\div 6{a}^{2}[\/latex]\r\n\r\n[latex]36{x}^{3}\\div \\left(-2{x}^{9}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{-18}{{x}^{6}}[\/latex]\r\n\r\n[latex]20{u}^{8}\\div \\left(-4{u}^{6}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{18{x}^{3}}{9{x}^{2}}[\/latex]\r\n\r\n2x<\/em>\r\n\r\n[latex]\\Large\\frac{36{y}^{9}}{4{y}^{7}}[\/latex]\r\n\r\n[latex]\\Large\\frac{-35{x}^{7}}{-42{x}^{13}}[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{6{x}^{6}}[\/latex]\r\n\r\n[latex]\\Large\\frac{18{x}^{5}}{-27{x}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{6{r}^{2}}{{s}^{8}}[\/latex]\r\n\r\n[latex]\\Large\\frac{24{p}^{7}q}{6{p}^{2}{q}^{5}}[\/latex]\r\n\r\n[latex]\\Large\\frac{8m{n}^{10}}{64m{n}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{n}^{6}}{8}[\/latex]\r\n\r\n[latex]\\Large\\frac{10{a}^{4}b}{50{a}^{2}{b}^{6}}[\/latex]\r\n\r\n[latex]\\Large\\frac{-12{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}}[\/latex]\r\n\r\n[latex]-\\Large\\frac{4{y}^{6}}{5{x}^{2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{48{x}^{11}{y}^{9}{z}^{3}}{36{x}^{6}{y}^{8}{z}^{5}}[\/latex]\r\n\r\n[latex]\\Large\\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}}[\/latex]\r\n\r\n[latex]\\Large\\frac{4z}{3{x}^{2}{y}^{3}}[\/latex]\r\n\r\n[latex]\\Large\\frac{\\left(10{u}^{2}v\\right)\\left(4{u}^{3}{v}^{6}\\right)}{5{u}^{9}{v}^{2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{\\left(6{m}^{2}n\\right)\\left(5{m}^{4}{n}^{3}\\right)}{3{m}^{10}{n}^{2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{10{n}^{2}}{{m}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{\\left(6{a}^{4}{b}^{3}\\right)\\left(4a{b}^{5}\\right)}{\\left(12{a}^{8}b\\right)\\left({a}^{3}b\\right)}[\/latex]\r\n\r\n[latex]\\Large\\frac{\\left(4{u}^{5}{v}^{4}\\right)\\left(15{u}^{8}v\\right)}{\\left(12{u}^{3}v\\right)\\left({u}^{6}v\\right)}[\/latex]\r\n\r\n5u<\/em>4<\/sup>v<\/em>3<\/sup>\r\n

        Mixed Practice<\/h2>\r\n\u24d0 [latex]24{a}^{5}+2{a}^{5}[\/latex]\r\n\u24d1 [latex]24{a}^{5}-2{a}^{5}[\/latex]\r\n\u24d2 [latex]24{a}^{5}\\cdot 2{a}^{5}[\/latex]\r\n\u24d3 [latex]24{a}^{5}\\div 2{a}^{5}[\/latex]\r\n\r\n\u24d0 [latex]15{n}^{10}+3{n}^{10}[\/latex]\r\n\u24d1 [latex]15{n}^{10}-3{n}^{10}[\/latex]\r\n\u24d2 [latex]15{n}^{10}\\cdot 3{n}^{10}[\/latex]\r\n\u24d3 [latex]15{n}^{10}\\div 3{n}^{10}[\/latex]\r\n\r\n\u24d0 [latex]18{n}^{10}[\/latex]\r\n\u24d1 [latex]12{n}^{10}[\/latex]\r\n\u24d2 [latex]45{n}^{20}[\/latex]\r\n\u24d3 [latex]5[\/latex]\r\n\r\n\u24d0 [latex]{p}^{4}\\cdot {p}^{6}[\/latex]\r\n\u24d1 [latex]{\\left({p}^{4}\\right)}^{6}[\/latex]\r\n\r\n\u24d0 [latex]{q}^{5}\\cdot {q}^{3}[\/latex]\r\n\u24d1 [latex]{\\left({q}^{5}\\right)}^{3}[\/latex]\r\n\r\n\u24d0 [latex]{q}^{8}[\/latex]\r\n\u24d1 [latex]{q}^{15}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{{y}^{3}}{y}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{y}{{y}^{3}}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{{z}^{6}}{{z}^{5}}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{{z}^{5}}{{z}^{6}}[\/latex]\r\n\r\n\u24d0 [latex]z[\/latex]\r\n\u24d1 [latex]\\Large\\frac{1}{z}[\/latex]\r\n\r\n[latex]\\left(8{x}^{5}\\right)\\left(9x\\right)\\div 6{x}^{3}[\/latex]\r\n\r\n[latex]\\left(4{y}^{5}\\right)\\left(12{y}^{7}\\right)\\div 8{y}^{2}[\/latex]\r\n\r\n[latex]6{y}^{6}[\/latex]\r\n\r\n[latex]\\Large\\frac{27{a}^{7}}{3{a}^{3}}\\normalsize +\\Large\\frac{54{a}^{9}}{9{a}^{5}}[\/latex]\r\n\r\n[latex]\\Large\\frac{32{c}^{11}}{4{c}^{5}}\\normalsize +\\Large\\frac{42{c}^{9}}{6{c}^{3}}[\/latex]\r\n\r\n[latex]15{c}^{6}[\/latex]\r\n\r\n[latex]\\Large\\frac{32{y}^{5}}{8{y}^{2}}\\normalsize -\\Large\\frac{60{y}^{10}}{5{y}^{7}}[\/latex]\r\n\r\n[latex]\\Large\\frac{48{x}^{6}}{6{x}^{4}}\\normalsize -\\Large\\frac{35{x}^{9}}{7{x}^{7}}[\/latex]\r\n\r\n[latex]3{x}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}\\normalsize -\\Large\\frac{72{r}^{2}{s}^{2}}{6s}[\/latex]\r\n\r\n[latex]\\Large\\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}\\normalsize -\\Large\\frac{45{y}^{2}{z}^{2}}{5y}[\/latex]\r\n\r\n[latex]y{z}^{2}[\/latex]\r\n

        Everyday Math<\/h2>\r\nMemory<\/strong> One megabyte is approximately [latex]{10}^{6}[\/latex] bytes. One gigabyte is approximately [latex]{10}^{9}[\/latex] bytes. How many megabytes are in one gigabyte?\r\n\r\nMemory<\/strong> One megabyte is approximately [latex]{10}^{6}[\/latex] bytes. One terabyte is approximately [latex]{10}^{12}[\/latex] bytes. How many megabytes are in one terabyte?\r\n\r\n1,000,000\r\n

        Writing Exercises<\/h2>\r\nVic thinks the quotient [latex]\\Large\\frac{{x}^{20}}{{x}^{4}}[\/latex] simplifies to [latex]{x}^{5}[\/latex]. What is wrong with his reasoning?\r\n\r\nMai simplifies the quotient [latex]\\Large\\frac{{y}^{3}}{y}[\/latex] by writing [latex]\\Large\\frac{{\\overline{)y}}^{3}}{\\overline{)y}}=3[\/latex]. What is wrong with her reasoning?\r\n\r\nAnswers will vary.\r\n\r\nWhen Dimple simplified [latex]-{3}^{0}[\/latex] and [latex]{\\left(-3\\right)}^{0}[\/latex] she got the same answer. Explain how using the Order of Operations correctly gives different answers.\r\n\r\nRoxie thinks [latex]{n}^{0}[\/latex] simplifies to [latex]0[\/latex]. What would you say to convince Roxie she is wrong?\r\n\r\nAnswers will vary.\r\n

        Self Check<\/h2>\r\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\r\n\r\n\".\"\r\n\u24d1 On a scale of 1\u201310, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?\r\n

        Practice Makes Perfect<\/h2>\r\nUse the Definition of a Negative Exponent<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{5}^{-3}[\/latex]\r\n\r\n[latex]{8}^{-2}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{64}[\/latex]\r\n\r\n[latex]{3}^{-4}[\/latex]\r\n\r\n[latex]{2}^{-5}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{32}[\/latex]\r\n\r\n[latex]{7}^{-1}[\/latex]\r\n\r\n[latex]{10}^{-1}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{10}[\/latex]\r\n\r\n[latex]{2}^{-3}+{2}^{-2}[\/latex]\r\n\r\n[latex]{3}^{-2}+{3}^{-1}[\/latex]\r\n\r\n[latex]\\Large\\frac{4}{9}[\/latex]\r\n\r\n[latex]{3}^{-1}+{4}^{-1}[\/latex]\r\n\r\n[latex]{10}^{-1}+{2}^{-1}[\/latex]\r\n\r\n[latex]\\Large\\frac{3}{5}[\/latex]\r\n\r\n[latex]{10}^{0}-{10}^{-1}+{10}^{-2}[\/latex]\r\n\r\n[latex]{2}^{0}-{2}^{-1}+{2}^{-2}[\/latex]\r\n\r\n[latex]\\Large\\frac{3}{4}[\/latex]\r\n\r\n\u24d0 [latex]{\\left(-6\\right)}^{-2}[\/latex]\r\n\u24d1 [latex]-{6}^{-2}[\/latex]\r\n\r\n\u24d0 [latex]{\\left(-8\\right)}^{-2}[\/latex]\r\n\u24d1 [latex]-{8}^{-2}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{1}{64}[\/latex]\r\n\u24d1 [latex]-\\Large\\frac{1}{64}[\/latex]\r\n\r\n\u24d0 [latex]{\\left(-10\\right)}^{-4}[\/latex]\r\n\u24d1 [latex]-{10}^{-4}[\/latex]\r\n\r\n\u24d0 [latex]{\\left(-4\\right)}^{-6}[\/latex]\r\n\u24d1 [latex]-{4}^{-6}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{1}{4096}[\/latex]\r\n\u24d1 [latex]-\\Large\\frac{1}{4096}[\/latex]\r\n\r\n\u24d0 [latex]5\\cdot {2}^{-1}[\/latex]\r\n\u24d1 [latex]{\\left(5\\cdot 2\\right)}^{-1}[\/latex]\r\n\r\n\u24d0 [latex]10\\cdot {3}^{-1}[\/latex]\r\n\u24d1 [latex]{\\left(10\\cdot 3\\right)}^{-1}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{10}{3}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{1}{30}[\/latex]\r\n\r\n\u24d0 [latex]4\\cdot {10}^{-3}[\/latex]\r\n\u24d1 [latex]{\\left(4\\cdot 10\\right)}^{-3}[\/latex]\r\n\r\n\u24d0 [latex]3\\cdot {5}^{-2}[\/latex]\r\n\u24d1 [latex]{\\left(3\\cdot 5\\right)}^{-2}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{3}{25}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{1}{225}[\/latex]\r\n\r\n[latex]{n}^{-4}[\/latex]\r\n\r\n[latex]{p}^{-3}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{p}^{3}}[\/latex]\r\n\r\n[latex]{c}^{-10}[\/latex]\r\n\r\n[latex]{m}^{-5}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{m}^{5}}[\/latex]\r\n\r\n\u24d0 [latex]4{x}^{-1}[\/latex]\r\n\u24d1 [latex]{\\left(4x\\right)}^{-1}[\/latex]\r\n\u24d2 [latex]{\\left(-4x\\right)}^{-1}[\/latex]\r\n\r\n\u24d0 [latex]3{q}^{-1}[\/latex]\r\n\u24d1 [latex]{\\left(3q\\right)}^{-1}[\/latex]\r\n\u24d2 [latex]{\\left(-3q\\right)}^{-1}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{3}{q}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{1}{3q}[\/latex]\r\n\u24d2 [latex]-\\Large\\frac{1}{3q}[\/latex]\r\n\r\n\u24d0 [latex]6{m}^{-1}[\/latex]\r\n\u24d1 [latex]{\\left(6m\\right)}^{-1}[\/latex]\r\n\u24d2 [latex]{\\left(-6m\\right)}^{-1}[\/latex]\r\n\r\n\u24d0 [latex]10{k}^{-1}[\/latex]\r\n\u24d1 [latex]{\\left(10k\\right)}^{-1}[\/latex]\r\n\u24d2 [latex]{\\left(-10k\\right)}^{-1}[\/latex]\r\n\r\n\u24d0 [latex]\\Large\\frac{10}{k}[\/latex]\r\n\u24d1 [latex]\\Large\\frac{1}{10k}[\/latex]\r\n\u24d2 [latex]-\\Large\\frac{1}{10k}[\/latex]\r\n\r\nSimplify Expressions with Integer Exponents<\/strong>\r\nIn the following exercises, simplify.<\/em>\r\n\r\n[latex]{p}^{-4}\\cdot {p}^{8}[\/latex]\r\n\r\n[latex]{r}^{-2}\\cdot {r}^{5}[\/latex]\r\n\r\nr<\/em>3<\/sup>\r\n\r\n[latex]{n}^{-10}\\cdot {n}^{2}[\/latex]\r\n\r\n[latex]{q}^{-8}\\cdot {q}^{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{q}^{5}}[\/latex]\r\n\r\n[latex]{k}^{-3}\\cdot {k}^{-2}[\/latex]\r\n\r\n[latex]{z}^{-6}\\cdot {z}^{-2}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{z}^{8}}[\/latex]\r\n\r\n[latex]a\\cdot {a}^{-4}[\/latex]\r\n\r\n[latex]m\\cdot {m}^{-2}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{m}[\/latex]\r\n\r\n[latex]{p}^{5}\\cdot {p}^{-2}\\cdot {p}^{-4}[\/latex]\r\n\r\n[latex]{x}^{4}\\cdot {x}^{-2}\\cdot {x}^{-3}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{x}[\/latex]\r\n\r\n[latex]{a}^{3}{b}^{-3}[\/latex]\r\n\r\n[latex]{u}^{2}{v}^{-2}[\/latex]\r\n\r\n[latex]\\Large\\frac{{u}^{2}}{{v}^{2}}[\/latex]\r\n\r\n[latex]\\left({x}^{5}{y}^{-1}\\right)\\left({x}^{-10}{y}^{-3}\\right)[\/latex]\r\n\r\n[latex]\\left({a}^{3}{b}^{-3}\\right)\\left({a}^{-5}{b}^{-1}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{a}^{2}{b}^{4}}[\/latex]\r\n\r\n[latex]\\left(u{v}^{-2}\\right)\\left({u}^{-5}{v}^{-4}\\right)[\/latex]\r\n\r\n[latex]\\left(p{q}^{-4}\\right)\\left({p}^{-6}{q}^{-3}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{p}^{5}{q}^{7}}[\/latex]\r\n\r\n[latex]\\left(-2{r}^{-3}{s}^{9}\\right)\\left(6{r}^{4}{s}^{-5}\\right)[\/latex]\r\n\r\n[latex]\\left(-3{p}^{-5}{q}^{8}\\right)\\left(7{p}^{2}{q}^{-3}\\right)[\/latex]\r\n\r\n[latex]-\\Large\\frac{21{q}^{5}}{{p}^{3}}[\/latex]\r\n\r\n[latex]\\left(-6{m}^{-8}{n}^{-5}\\right)\\left(-9{m}^{4}{n}^{2}\\right)[\/latex]\r\n\r\n[latex]\\left(-8{a}^{-5}{b}^{-4}\\right)\\left(-4{a}^{2}{b}^{3}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{32}{{a}^{3}b}[\/latex]\r\n\r\n[latex]{\\left({a}^{3}\\right)}^{-3}[\/latex]\r\n\r\n[latex]{\\left({q}^{10}\\right)}^{-10}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{q}^{100}}[\/latex]\r\n\r\n[latex]{\\left({n}^{2}\\right)}^{-1}[\/latex]\r\n\r\n[latex]{\\left({x}^{4}\\right)}^{-1}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{x}^{4}}[\/latex]\r\n\r\n[latex]{\\left({y}^{-5}\\right)}^{4}[\/latex]\r\n\r\n[latex]{\\left({p}^{-3}\\right)}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{y}^{6}}[\/latex]\r\n\r\n[latex]{\\left({q}^{-5}\\right)}^{-2}[\/latex]\r\n\r\n[latex]{\\left({m}^{-2}\\right)}^{-3}[\/latex]\r\n\r\nm<\/em>6<\/sup>\r\n\r\n[latex]{\\left(4{y}^{-3}\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left(3{q}^{-5}\\right)}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{9}{{q}^{10}}[\/latex]\r\n\r\n[latex]{\\left(10{p}^{-2}\\right)}^{-5}[\/latex]\r\n\r\n[latex]{\\left(2{n}^{-3}\\right)}^{-6}[\/latex]\r\n\r\n[latex]\\Large\\frac{{n}^{18}}{64}[\/latex]\r\n\r\n[latex]\\Large\\frac{{u}^{9}}{{u}^{-2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{b}^{5}}{{b}^{-3}}[\/latex]\r\n\r\nb<\/em>8<\/sup>\r\n\r\n[latex]\\Large\\frac{{x}^{-6}}{{x}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{m}^{5}}{{m}^{-2}}[\/latex]\r\n\r\nm<\/em>7<\/sup>\r\n\r\n[latex]\\Large\\frac{{q}^{3}}{{q}^{12}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{r}^{6}}{{r}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{r}^{3}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{n}^{-4}}{{n}^{-10}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{p}^{-3}}{{p}^{-6}}[\/latex]\r\n\r\np<\/em>3<\/sup>\r\n\r\nConvert from Decimal Notation to Scientific Notation<\/strong>\r\nIn the following exercises, write each number in scientific notation.\r\n\r\n45,000\r\n\r\n280,000\r\n\r\n2.8 \u00d7 105<\/sup>\r\n\r\n8,750,000\r\n\r\n1,290,000\r\n\r\n1.29 \u00d7 106<\/sup>\r\n\r\n0.036\r\n\r\n0.041\r\n\r\n4.1 \u00d7 10\u22122<\/sup>\r\n\r\n0.00000924\r\n\r\n0.0000103\r\n\r\n1.03 \u00d7 10\u22125<\/sup>\r\n\r\nThe population of the United States on July 4, 2010 was almost [latex]310,000,000[\/latex].\r\n\r\nThe population of the world on July 4, 2010 was more than [latex]6,850,000,000[\/latex].\r\n\r\n6.85 \u00d7 109<\/sup>\r\n\r\nThe average width of a human hair is [latex]0.0018[\/latex] centimeters.\r\n\r\nThe probability of winning the [latex]2010[\/latex] Megamillions lottery is about [latex]0.0000000057[\/latex].\r\n\r\n5.7 \u00d7 10\u22129<\/sup>\r\n\r\nConvert Scientific Notation to Decimal Form<\/strong>\r\nIn the following exercises, convert each number to decimal form.\r\n\r\n[latex]4.1\\times {10}^{2}[\/latex]\r\n\r\n[latex]8.3\\times {10}^{2}[\/latex]\r\n\r\n830\r\n\r\n[latex]5.5\\times {10}^{8}[\/latex]\r\n\r\n[latex]1.6\\times {10}^{10}[\/latex]\r\n\r\n16,000,000,000\r\n\r\n[latex]3.5\\times {10}^{-2}[\/latex]\r\n\r\n[latex]2.8\\times {10}^{-2}[\/latex]\r\n\r\n0.028\r\n\r\n[latex]1.93\\times {10}^{-5}[\/latex]\r\n\r\n[latex]6.15\\times {10}^{-8}[\/latex]\r\n\r\n0.0000000615\r\n\r\nIn 2010, the number of Facebook users each day who changed their status to \u2018engaged\u2019 was [latex]2\\times {10}^{4}[\/latex].\r\n\r\nAt the start of 2012, the US federal budget had a deficit of more than [latex]{$1.5}\\times {10}^{13}[\/latex].\r\n\r\n$15,000,000,000,000\r\n\r\nThe concentration of carbon dioxide in the atmosphere is [latex]3.9\\times {10}^{-4}[\/latex].\r\n\r\nThe width of a proton is [latex]1\\times {10}^{-5}[\/latex] of the width of an atom.\r\n\r\n0.00001\r\n\r\nMultiply and Divide Using Scientific Notation<\/strong>\r\nIn the following exercises, multiply or divide and write your answer in decimal form.\r\n\r\n[latex]\\left(2\\times {10}^{5}\\right)\\left(2\\times {10}^{-9}\\right)[\/latex]\r\n\r\n[latex]\\left(3\\times {10}^{2}\\right)\\left(1\\times {10}^{-5}\\right)[\/latex]\r\n\r\n0.003\r\n\r\n[latex]\\left(1.6\\times {10}^{-2}\\right)\\left(5.2\\times {10}^{-6}\\right)[\/latex]\r\n\r\n[latex]\\left(2.1\\times {10}^{-4}\\right)\\left(3.5\\times {10}^{-2}\\right)[\/latex]\r\n\r\n0.00000735\r\n\r\n[latex]\\Large\\frac{6\\times {10}^{4}}{3\\times {10}^{-2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{8\\times {10}^{6}}{4\\times {10}^{-1}}[\/latex]\r\n\r\n200,000\r\n\r\n[latex]\\Large\\frac{7\\times {10}^{-2}}{1\\times {10}^{-8}}[\/latex]\r\n\r\n[latex]\\Large\\frac{5\\times {10}^{-3}}{1\\times {10}^{-10}}[\/latex]\r\n\r\n50,000,000\r\n

        Everyday Math<\/h2>\r\nCalories<\/strong> In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by [latex]1.5[\/latex] trillion calories by the end of 2015.\r\n
          \r\n \t
        1. \u24d0 Write [latex]1.5[\/latex] trillion in decimal notation.<\/li>\r\n \t
        2. \u24d1 Write [latex]1.5[\/latex] trillion in scientific notation.<\/li>\r\n<\/ol>\r\nLength of a year<\/strong> The difference between the calendar year and the astronomical year is [latex]0.000125[\/latex] day.\r\n
            \r\n \t
          1. \u24d0 Write this number in scientific notation.<\/li>\r\n \t
          2. \u24d1 How many years does it take for the difference to become 1 day?<\/li>\r\n<\/ol>\r\n
              \r\n \t
            1. \u24d0 1.25 \u00d7 10\u22124<\/sup><\/li>\r\n \t
            2. \u24d0 8,000<\/li>\r\n<\/ol>\r\nCalculator display<\/strong> Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator\u2019s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided [latex]1[\/latex] by [latex]2,598,960[\/latex] and saw the answer [latex]3.848\\times {10}^{-7}[\/latex]. Write the number in decimal notation.\r\n\r\nCalculator display<\/strong> Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator\u2019s display. To find the number of ways Barbara could make a collage with [latex]6[\/latex] of her [latex]50[\/latex] favorite photographs, she multiplied [latex]50\\cdot 49\\cdot 48\\cdot 47\\cdot 46\\cdot 45[\/latex]. Her calculator gave the answer [latex]1.1441304\\times {10}^{10}[\/latex]. Write the number in decimal notation.\r\n\r\n11,441,304,000\r\n

              Writing Exercises<\/h2>\r\n
                \r\n \t
              1. \u24d0 Explain the meaning of the exponent in the expression [latex]{2}^{3}[\/latex].<\/li>\r\n \t
              2. \u24d1 Explain the meaning of the exponent in the expression [latex]{2}^{-3}[\/latex]<\/li>\r\n<\/ol>\r\nWhen you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?\r\n\r\nAnswers will vary.\r\n

                Self Check<\/h2>\r\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\r\n\r\n\".\"\r\n\u24d1 After looking at the checklist, do you think you are well prepared for the next section? Why or why not?\r\n

                <\/h2>\r\n

                Practice Makes Perfect<\/h2>\r\nFind the Greatest Common Factor of Two or More Expressions<\/strong>\r\nIn the following exercises, find the greatest common factor.\r\n\r\n[latex]40,56[\/latex]\r\n\r\n[latex]45,75[\/latex]\r\n\r\n15\r\n\r\n[latex]72,162[\/latex]\r\n\r\n[latex]150,275[\/latex]\r\n\r\n25\r\n\r\n[latex]3x,12[\/latex]\r\n\r\n[latex]4y,28[\/latex]\r\n\r\n4\r\n\r\n[latex]10a,50[\/latex]\r\n\r\n[latex]5b,30[\/latex]\r\n\r\n5\r\n\r\n[latex]16y,24{y}^{2}[\/latex]\r\n\r\n[latex]9x,15{x}^{2}[\/latex]\r\n\r\n3x<\/em>\r\n\r\n[latex]18{m}^{3},36{m}^{2}[\/latex]\r\n\r\n[latex]12{p}^{4},48{p}^{3}[\/latex]\r\n\r\n12p<\/em>3<\/sup>\r\n\r\n[latex]10x,25{x}^{2},15{x}^{3}[\/latex]\r\n\r\n[latex]18a,6{a}^{2},22{a}^{3}[\/latex]\r\n\r\n2a<\/em>\r\n\r\n[latex]24u,6{u}^{2},30{u}^{3}[\/latex]\r\n\r\n[latex]40y,10{y}^{2},90{y}^{3}[\/latex]\r\n\r\n10y<\/em>\r\n\r\n[latex]15{a}^{4},9{a}^{5},21{a}^{6}[\/latex]\r\n\r\n[latex]35{x}^{3},10{x}^{4},5{x}^{5}[\/latex]\r\n\r\n5x<\/em>3<\/sup>\r\n\r\n[latex]27{y}^{2},45{y}^{3},9{y}^{4}[\/latex]\r\n\r\n[latex]14{b}^{2},35{b}^{3},63{b}^{4}[\/latex]\r\n\r\n7b<\/em>2<\/sup>\r\n\r\nFactor the Greatest Common Factor from a Polynomial<\/strong>\r\nIn the following exercises, factor the greatest common factor from each polynomial.\r\n\r\n[latex]2x+8[\/latex]\r\n\r\n[latex]5y+15[\/latex]\r\n\r\n5(y<\/em> + 3)\r\n\r\n[latex]3a - 24[\/latex]\r\n\r\n[latex]4b - 20[\/latex]\r\n\r\n4(b<\/em> \u2212 5)\r\n\r\n[latex]9y - 9[\/latex]\r\n\r\n[latex]7x - 7[\/latex]\r\n\r\n7(x<\/em> \u2212 1)\r\n\r\n[latex]5{m}^{2}+20m+35[\/latex]\r\n\r\n[latex]3{n}^{2}+21n+12[\/latex]\r\n\r\n3(n<\/em>2<\/sup> + 7n<\/em> + 4)\r\n\r\n[latex]8{p}^{2}+32p+48[\/latex]\r\n\r\n[latex]6{q}^{2}+30q+42[\/latex]\r\n\r\n6(q<\/em>2<\/sup> + 5q<\/em> + 7)\r\n\r\n[latex]8{q}^{2}+15q[\/latex]\r\n\r\n[latex]9{c}^{2}+22c[\/latex]\r\n\r\nc<\/em>(9c<\/em> + 22)\r\n\r\n[latex]13{k}^{2}+5k[\/latex]\r\n\r\n[latex]17{x}^{2}+7x[\/latex]\r\n\r\nx<\/em>(17x<\/em> + 7)\r\n\r\n[latex]5{c}^{2}+9c[\/latex]\r\n\r\n[latex]4{q}^{2}+7q[\/latex]\r\n\r\nq<\/em>(4q<\/em> + 7)\r\n\r\n[latex]5{p}^{2}+25p[\/latex]\r\n\r\n[latex]3{r}^{2}+27r[\/latex]\r\n\r\n3r<\/em>(r<\/em> + 9)\r\n\r\n[latex]24{q}^{2}-12q[\/latex]\r\n\r\n[latex]30{u}^{2}-10u[\/latex]\r\n\r\n10u<\/em>(3u<\/em> \u2212 1)\r\n\r\n[latex]yz+4z[\/latex]\r\n\r\n[latex]ab+8b[\/latex]\r\n\r\nb<\/em>(a<\/em> + 8)\r\n\r\n[latex]60x - 6{x}^{3}[\/latex]\r\n\r\n[latex]55y - 11{y}^{4}[\/latex]\r\n\r\n11y<\/em>(5 \u2212 y<\/em>3<\/sup>)\r\n\r\n[latex]48{r}^{4}-12{r}^{3}[\/latex]\r\n\r\n[latex]45{c}^{3}-15{c}^{2}[\/latex]\r\n\r\n15c<\/em>2<\/sup>(3c<\/em> \u2212 1)\r\n\r\n[latex]4{a}^{3}-4a{b}^{2}[\/latex]\r\n\r\n[latex]6{c}^{3}-6c{d}^{2}[\/latex]\r\n\r\n6c<\/em>(c<\/em>2<\/sup> \u2212 d<\/em>2<\/sup>)\r\n\r\n[latex]30{u}^{3}+80{u}^{2}[\/latex]\r\n\r\n[latex]48{x}^{3}+72{x}^{2}[\/latex]\r\n\r\n24x<\/em>2<\/sup>(2x<\/em> + 3)\r\n\r\n[latex]120{y}^{6}+48{y}^{4}[\/latex]\r\n\r\n[latex]144{a}^{6}+90{a}^{3}[\/latex]\r\n\r\n18a<\/em>3<\/sup>(8a<\/em>3<\/sup> + 5)\r\n\r\n[latex]4{q}^{2}+24q+28[\/latex]\r\n\r\n[latex]10{y}^{2}+50y+40[\/latex]\r\n\r\n10(y<\/em>2<\/sup> + 5y<\/em> + 4)\r\n\r\n[latex]15{z}^{2}-30z - 90[\/latex]\r\n\r\n[latex]12{u}^{2}-36u - 108[\/latex]\r\n\r\n12(u<\/em>2<\/sup> \u2212 3u<\/em> \u2212 9)\r\n\r\n[latex]3{a}^{4}-24{a}^{3}+18{a}^{2}[\/latex]\r\n\r\n[latex]5{p}^{4}-20{p}^{3}-15{p}^{2}[\/latex]\r\n\r\n5p<\/em>2<\/sup>(p<\/em>2<\/sup> \u2212 4p<\/em> \u2212 3)\r\n\r\n[latex]11{x}^{6}+44{x}^{5}-121{x}^{4}[\/latex]\r\n\r\n[latex]8{c}^{5}+40{c}^{4}-56{c}^{3}[\/latex]\r\n\r\n8c<\/em>3<\/sup>(c<\/em>2<\/sup> + 5c<\/em> \u2212 7)\r\n\r\n[latex]-3n - 24[\/latex]\r\n\r\n[latex]-7p - 84[\/latex]\r\n\r\n\u22127(p<\/em> + 12)\r\n\r\n[latex]-15{a}^{2}-40a[\/latex]\r\n\r\n[latex]-18{b}^{2}-66b[\/latex]\r\n\r\n\u22126b<\/em>(3b<\/em> + 11)\r\n\r\n[latex]-10{y}^{3}+60{y}^{2}[\/latex]\r\n\r\n[latex]-8{a}^{3}+32{a}^{2}[\/latex]\r\n\r\n\u22128a<\/em>2<\/sup>(a<\/em> \u2212 4)\r\n\r\n[latex]-4{u}^{5}+56{u}^{3}[\/latex]\r\n\r\n[latex]-9{b}^{5}+63{b}^{3}[\/latex]\r\n\r\n\u22129b<\/em>3<\/sup>(b<\/em>2<\/sup> \u2212 7)\r\n

                Everyday Math<\/h2>\r\nRevenue<\/strong> A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of [latex]p[\/latex] dollars each is given by the polynomial [latex]-5{p}^{2}+150p[\/latex]. Factor the greatest common factor from this polynomial.\r\n\r\nHeight of a baseball<\/strong> The height of a baseball hit with velocity [latex]80[\/latex] feet\/second at [latex]4[\/latex] feet above ground level is [latex]-16{t}^{2}+80t+4[\/latex], with [latex]t=[\/latex] the number of seconds since it was hit. Factor the greatest common factor from this polynomial.\r\n\r\n\u22124(4t<\/em>2<\/sup> \u2212 20t<\/em> \u2212 1)\r\n

                Writing Exercises<\/h2>\r\nThe greatest common factor of [latex]36[\/latex] and [latex]60[\/latex] is [latex]12[\/latex]. Explain what this means.\r\n\r\nWhat is the GCF of [latex]{y}^{4}[\/latex] , [latex]{y}^{5}[\/latex] , and [latex]{y}^{10}[\/latex] ? Write a general rule that tells how to find the GCF of [latex]{y}^{\\text{a}}[\/latex] , [latex]{y}^{\\text{b}}[\/latex] , and [latex]{y}^{\\text{c}}[\/latex] .\r\n

                Self Check<\/h2>\r\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\r\n\r\n\".\"\r\n\u24d1 Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?\r\n

                Chapter Review Exercises<\/h1>\r\n

                Add and Subtract Polynomials<\/h2>\r\nIdentify Polynomials, Monomials, Binomials and Trinomials<\/strong>\r\nIn the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.\r\n\r\n[latex]{y}^{2}+8y - 20[\/latex]\r\n\r\ntrinomial\r\n\r\n[latex]-6{a}^{4}[\/latex]\r\n\r\n[latex]9{x}^{3}-1[\/latex]\r\n\r\nbinomial\r\n\r\n[latex]{n}^{3}-3{n}^{2}+3n - 1[\/latex]\r\n\r\nDetermine the Degree of Polynomials<\/strong>\r\nIn the following exercises, determine the degree of each polynomial.\r\n\r\n[latex]16{x}^{2}-40x - 25[\/latex]\r\n\r\n2\r\n\r\n[latex]5m+9[\/latex]\r\n\r\n[latex]-15[\/latex]\r\n\r\n0\r\n\r\n[latex]{y}^{2}+6{y}^{3}+9{y}^{4}[\/latex]\r\n\r\nAdd and Subtract Monomials<\/strong>\r\nIn the following exercises, add or subtract the monomials.\r\n\r\n[latex]4p+11p[\/latex]\r\n\r\n15p<\/em>\r\n\r\n[latex]-8{y}^{3}-5{y}^{3}[\/latex]\r\n\r\nAdd [latex]4{n}^{5},\\text{-}{n}^{5},-6{n}^{5}[\/latex]\r\n\r\n\u22123n<\/em>5<\/sup>\r\n\r\nSubtract [latex]10{x}^{2}[\/latex] from [latex]3{x}^{2}[\/latex]\r\n\r\nAdd and Subtract Polynomials<\/strong>\r\nIn the following exercises, add or subtract the polynomials.\r\n\r\n[latex]\\left(4{a}^{2}+9a - 11\\right)+\\left(6{a}^{2}-5a+10\\right)[\/latex]\r\n\r\n10a<\/em>2<\/sup> + 4a<\/em> \u2212 1\r\n\r\n[latex]\\left(8{m}^{2}+12m - 5\\right)-\\left(2{m}^{2}-7m - 1\\right)[\/latex]\r\n\r\n[latex]\\left({y}^{2}-3y+12\\right)+\\left(5{y}^{2}-9\\right)[\/latex]\r\n\r\n6y<\/em>2<\/sup> \u2212 3y<\/em> + 3\r\n\r\n[latex]\\left(5{u}^{2}+8u\\right)-\\left(4u - 7\\right)[\/latex]\r\n\r\nFind the sum of [latex]8{q}^{3}-27[\/latex] and [latex]{q}^{2}+6q - 2[\/latex]\r\n\r\n8q<\/em>3<\/sup> + q<\/em>2<\/sup> + 6q<\/em> \u2212 29\r\n\r\nFind the difference of [latex]{x}^{2}+6x+8[\/latex] and [latex]{x}^{2}-8x+15[\/latex]\r\n\r\nEvaluate a Polynomial for a Given Value of the Variable<\/strong>\r\nIn the following exercises, evaluate each polynomial for the given value.\r\n\r\n[latex]200x-\\Large\\frac{1}{5}\\normalsize{x}^{2}[\/latex] when [latex]x=5[\/latex]\r\n\r\n995\r\n\r\n[latex]200x-\\Large\\frac{1}{5}\\normalsize{x}^{2}[\/latex] when [latex]x=0[\/latex]\r\n\r\n[latex]200x-\\Large\\frac{1}{5}\\normalsize{x}^{2}[\/latex] when [latex]x=15[\/latex]\r\n\r\n2,955\r\n\r\n[latex]5+40x-\\Large\\frac{1}{2}\\normalsize{x}^{2}[\/latex] when [latex]x=10[\/latex]\r\n\r\n[latex]5+40x-\\Large\\frac{1}{2}\\normalsize{x}^{2}[\/latex] when [latex]x=-4[\/latex]\r\n\r\n\u2212163\r\n\r\n[latex]5+40x-\\Large\\frac{1}{2}\\normalsize{x}^{2}[\/latex] when [latex]x=0[\/latex]\r\n\r\nA pair of glasses is dropped off a bridge [latex]640[\/latex] feet above a river. The polynomial [latex]-16{t}^{2}+640[\/latex] gives the height of the glasses [latex]t[\/latex] seconds after they were dropped. Find the height of the glasses when [latex]t=6[\/latex].\r\n\r\n64 feet\r\n\r\nThe fuel efficiency (in miles per gallon) of a bus going at a speed of [latex]x[\/latex] miles per hour is given by the polynomial [latex]-\\Large\\frac{1}{160}\\normalsize{x}^{2}+\\Large\\frac{1}{2}\\normalsize x[\/latex]. Find the fuel efficiency when [latex]x=20[\/latex] mph.\r\n

                Use Multiplication Properties of Exponents<\/h2>\r\nSimplify Expressions with Exponents<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{6}^{3}[\/latex]\r\n\r\n216\r\n\r\n[latex]{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{4}[\/latex]\r\n\r\n[latex]{\\left(-0.5\\right)}^{2}[\/latex]\r\n\r\n0.25\r\n\r\n[latex]-{3}^{2}[\/latex]\r\n\r\nSimplify Expressions Using the Product Property of Exponents<\/strong>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]{p}^{3}\\cdot {p}^{10}[\/latex]\r\n\r\np<\/em>13<\/sup>\r\n\r\n[latex]2\\cdot {2}^{6}[\/latex]\r\n\r\n[latex]a\\cdot {a}^{2}\\cdot {a}^{3}[\/latex]\r\n\r\na<\/em>6<\/sup>\r\n\r\n[latex]x\\cdot {x}^{8}[\/latex]\r\n\r\nSimplify Expressions Using the Power Property of Exponents<\/strong>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]{\\left({y}^{4}\\right)}^{3}[\/latex]\r\n\r\ny<\/em>12<\/sup>\r\n\r\n[latex]{\\left({r}^{3}\\right)}^{2}[\/latex]\r\n\r\n[latex]{\\left({3}^{2}\\right)}^{5}[\/latex]\r\n\r\n310<\/sup>\r\n\r\n[latex]{\\left({a}^{10}\\right)}^{y}[\/latex]\r\n\r\nSimplify Expressions Using the Product to a Power Property<\/strong>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]{\\left(8n\\right)}^{2}[\/latex]\r\n\r\n64n<\/em>2<\/sup>\r\n\r\n[latex]{\\left(-5x\\right)}^{3}[\/latex]\r\n\r\n[latex]{\\left(2ab\\right)}^{8}[\/latex]\r\n\r\n256a<\/em>8<\/sup>b<\/em>8<\/sup>\r\n\r\n[latex]{\\left(-10mnp\\right)}^{4}[\/latex]\r\n\r\nSimplify Expressions by Applying Several Properties<\/strong>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]{\\left(3{a}^{5}\\right)}^{3}[\/latex]\r\n\r\n27a<\/em>15<\/sup>\r\n\r\n[latex]{\\left(4y\\right)}^{2}\\left(8y\\right)[\/latex]\r\n\r\n[latex]{\\left({x}^{3}\\right)}^{5}{\\left({x}^{2}\\right)}^{3}[\/latex]\r\n\r\nx<\/em>21<\/sup>\r\n\r\n[latex]{\\left(5s{t}^{2}\\right)}^{3}{\\left(2{s}^{3}{t}^{4}\\right)}^{2}[\/latex]\r\n\r\nMultiply Monomials<\/strong>\r\nIn the following exercises, multiply the monomials.\r\n\r\n[latex]\\left(-6{p}^{4}\\right)\\left(9p\\right)[\/latex]\r\n\r\n\u221254p<\/em>5<\/sup>\r\n\r\n[latex]\\left(\\Large\\frac{1}{3}\\normalsize{c}^{2}\\right)\\left(30{c}^{8}\\right)[\/latex]\r\n\r\n[latex]\\left(8{x}^{2}{y}^{5}\\right)\\left(7x{y}^{6}\\right)[\/latex]\r\n\r\n56x<\/em>3<\/sup>y<\/em>11<\/sup>\r\n\r\n[latex]\\left(\\Large\\frac{2}{3}\\normalsize{m}^{3}{n}^{6}\\right)\\left(\\Large\\frac{1}{6}\\normalsize{m}^{4}{n}^{4}\\right)[\/latex]\r\n

                Multiply Polynomials<\/h2>\r\nMultiply a Polynomial by a Monomial<\/strong>\r\nIn the following exercises, multiply.\r\n\r\n[latex]7\\left(10-x\\right)[\/latex]\r\n\r\n70 \u2212 7x<\/em>\r\n\r\n[latex]{a}^{2}\\left({a}^{2}-9a - 36\\right)[\/latex]\r\n\r\n[latex]-5y\\left(125{y}^{3}-1\\right)[\/latex]\r\n\r\n\u2212625y<\/em>4<\/sup> + 5y<\/em>\r\n\r\n[latex]\\left(4n - 5\\right)\\left(2{n}^{3}\\right)[\/latex]\r\n\r\nMultiply a Binomial by a Binomial<\/strong>\r\nIn the following exercises, multiply the binomials using various methods.\r\n\r\n[latex]\\left(a+5\\right)\\left(a+2\\right)[\/latex]\r\n\r\na<\/em>2<\/sup> + 7a<\/em> + 10\r\n\r\n[latex]\\left(y - 4\\right)\\left(y+12\\right)[\/latex]\r\n\r\n[latex]\\left(3x+1\\right)\\left(2x - 7\\right)[\/latex]\r\n\r\n6x<\/em>2<\/sup> \u2212 19x<\/em> \u2212 7\r\n\r\n[latex]\\left(6p - 11\\right)\\left(3p - 10\\right)[\/latex]\r\n\r\n[latex]\\left(n+8\\right)\\left(n+1\\right)[\/latex]\r\n\r\nn<\/em>2<\/sup> + 9n<\/em> + 8\r\n\r\n[latex]\\left(k+6\\right)\\left(k - 9\\right)[\/latex]\r\n\r\n[latex]\\left(5u - 3\\right)\\left(u+8\\right)[\/latex]\r\n\r\n5u<\/em>2<\/sup> + 37u<\/em> \u2212 24\r\n\r\n[latex]\\left(2y - 9\\right)\\left(5y - 7\\right)[\/latex]\r\n\r\n[latex]\\left(p+4\\right)\\left(p+7\\right)[\/latex]\r\n\r\np<\/em>2<\/sup> + 11p<\/em> + 28\r\n\r\n[latex]\\left(x - 8\\right)\\left(x+9\\right)[\/latex]\r\n\r\n[latex]\\left(3c+1\\right)\\left(9c - 4\\right)[\/latex]\r\n\r\n27c<\/em>2<\/sup> \u2212 3c<\/em> \u2212 4\r\n\r\n[latex]\\left(10a - 1\\right)\\left(3a - 3\\right)[\/latex]\r\n\r\nMultiply a Trinomial by a Binomial<\/strong>\r\nIn the following exercises, multiply using any method.\r\n\r\n[latex]\\left(x+1\\right)\\left({x}^{2}-3x - 21\\right)[\/latex]\r\n\r\nx<\/em>3<\/sup> \u2212 2x<\/em>2<\/sup> \u2212 24x<\/em> \u2212 21\r\n\r\n[latex]\\left(5b - 2\\right)\\left(3{b}^{2}+b - 9\\right)[\/latex]\r\n\r\n[latex]\\left(m+6\\right)\\left({m}^{2}-7m - 30\\right)[\/latex]\r\n\r\nm<\/em>3<\/sup> \u2212 m<\/em>2<\/sup> \u2212 72m<\/em> \u2212 180\r\n\r\n[latex]\\left(4y - 1\\right)\\left(6{y}^{2}-12y+5\\right)[\/latex]\r\n

                Divide Monomials<\/h2>\r\nSimplify Expressions Using the Quotient Property of Exponents<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]\\Large\\frac{{2}^{8}}{{2}^{2}}[\/latex]\r\n\r\n26<\/sup>or<\/em> 64\r\n\r\n[latex]\\Large\\frac{{a}^{6}}{a}[\/latex]\r\n\r\n[latex]\\Large\\frac{{n}^{3}}{{n}^{12}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{n}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{x}{{x}^{5}}[\/latex]\r\n\r\nSimplify Expressions with Zero Exponents<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{3}^{0}[\/latex]\r\n\r\n1\r\n\r\n[latex]{y}^{0}[\/latex]\r\n\r\n[latex]{\\left(14t\\right)}^{0}[\/latex]\r\n\r\n1\r\n\r\n[latex]12{a}^{0}-15{b}^{0}[\/latex]\r\n\r\nSimplify Expressions Using the Quotient to a Power Property<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{\\left(\\Large\\frac{3}{5}\\normalsize\\right)}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{9}{25}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{x}{2}\\normalsize\\right)}^{5}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{5m}{n}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]\\Large\\frac{125{m}^{3}}{{n}^{3}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{s}{10t}\\normalsize\\right)}^{2}[\/latex]\r\n\r\nSimplify Expressions by Applying Several Properties<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]\\Large\\frac{{\\left({a}^{3}\\right)}^{2}}{{a}^{4}}[\/latex]\r\n\r\na<\/em>2<\/sup>\r\n\r\n[latex]\\Large\\frac{{u}^{3}}{{u}^{2}\\normalsize\\cdot {u}^{4}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{x}{{x}^{9}}\\normalsize\\right)}^{5}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{x}^{40}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{{p}^{4}\\cdot {p}^{5}}{{p}^{3}}\\normalsize\\right)}^{2}[\/latex]\r\n\r\n[latex]\\Large\\frac{{\\left({n}^{5}\\right)}^{3}}{{\\left({n}^{2}\\right)}^{8}}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{n}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{5{s}^{2}}{4t}\\normalsize\\right)}^{3}[\/latex]\r\n\r\nDivide Monomials<\/strong>\r\nIn the following exercises, divide the monomials.\r\n\r\n[latex]72{p}^{12}\\div 8{p}^{3}[\/latex]\r\n\r\n9p<\/em>9<\/sup>\r\n\r\n[latex]-26{a}^{8}\\div \\left(2{a}^{2}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{45{y}^{6}}{-15{y}^{10}}[\/latex]\r\n\r\n[latex]-\\Large\\frac{3}{{y}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{-30{x}^{8}}{-36{x}^{9}}[\/latex]\r\n\r\n[latex]\\Large\\frac{28{a}^{9}b}{7{a}^{4}{b}^{3}}[\/latex]\r\n\r\n[latex]\\Large\\frac{4{a}^{5}}{{b}^{2}}[\/latex]\r\n\r\n[latex]\\Large\\frac{11{u}^{6}{v}^{3}}{55{u}^{2}{v}^{8}}[\/latex]\r\n\r\n[latex]\\Large\\frac{\\left(5{m}^{9}{n}^{3}\\right)\\left(8{m}^{3}{n}^{2}\\right)}{\\left(10m{n}^{4}\\right)\\left({m}^{2}{n}^{5}\\right)}[\/latex]\r\n\r\n[latex]\\Large\\frac{4{m}^{9}}{{n}^{4}}[\/latex]\r\n\r\n[latex]\\Large\\frac{42{r}^{2}{s}^{4}}{6r{s}^{3}}\\normalsize -\\Large\\frac{54r{s}^{2}}{9s}[\/latex]\r\n

                Integer Exponents and Scientific Notation<\/h2>\r\nUse the Definition of a Negative Exponent<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{6}^{-2}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{36}[\/latex]\r\n\r\n[latex]{\\left(-10\\right)}^{-3}[\/latex]\r\n\r\n[latex]5\\cdot {2}^{-4}[\/latex]\r\n\r\n[latex]\\Large\\frac{5}{16}[\/latex]\r\n\r\n[latex]{\\left(8n\\right)}^{-1}[\/latex]\r\n\r\nSimplify Expressions with Integer Exponents<\/strong>\r\nIn the following exercises, simplify.\r\n\r\n[latex]{x}^{-3}\\cdot {x}^{9}[\/latex]\r\n\r\nx<\/em>6<\/sup>\r\n\r\n[latex]{r}^{-5}\\cdot {r}^{-4}[\/latex]\r\n\r\n[latex]\\left(u{v}^{-3}\\right)\\left({u}^{-4}{v}^{-2}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{{u}^{3}{v}^{5}}[\/latex]\r\n\r\n[latex]{\\left({m}^{5}\\right)}^{-1}[\/latex]\r\n\r\n[latex]{\\left({k}^{-2}\\right)}^{-3}[\/latex]\r\n\r\nk<\/em>6<\/sup>\r\n\r\n[latex]\\Large\\frac{{q}^{4}}{{q}^{20}}[\/latex]\r\n\r\n[latex]\\Large\\frac{{b}^{8}}{{b}^{-2}}[\/latex]\r\n\r\nb<\/em>10<\/sup>\r\n\r\n[latex]\\Large\\frac{{n}^{-3}}{{n}^{-5}}[\/latex]\r\n\r\nConvert from Decimal Notation to Scientific Notation<\/strong>\r\nIn the following exercises, write each number in scientific notation.\r\n\r\n[latex]5,300,000[\/latex]\r\n\r\n5.3 \u00d7 106<\/sup>\r\n\r\n[latex]0.00814[\/latex]\r\n\r\nThe thickness of a piece of paper is about [latex]0.097[\/latex] millimeter.\r\n\r\n9.7 \u00d7 10\u22122<\/sup>\r\n\r\nAccording to www.cleanair.com, U.S. businesses use about [latex]21,000,000[\/latex] tons of paper per year.\r\n\r\nConvert Scientific Notation to Decimal Form<\/strong>\r\nIn the following exercises, convert each number to decimal form.\r\n\r\n[latex]2.9\\times {10}^{4}[\/latex]\r\n\r\n29,000\r\n\r\n[latex]1.5\\times {10}^{8}[\/latex]\r\n\r\n[latex]3.75\\times {10}^{-1}[\/latex]\r\n\r\n375\r\n\r\n[latex]9.413\\times {10}^{-5}[\/latex]\r\n\r\nMultiply and Divide Using Scientific Notation<\/strong>\r\nIn the following exercises, multiply and write your answer in decimal form.\r\n\r\n[latex]\\left(3\\times {10}^{7}\\right)\\left(2\\times {10}^{-4}\\right)[\/latex]\r\n\r\n6,000\r\n\r\n[latex]\\left(1.5\\times {10}^{-3}\\right)\\left(4.8\\times {10}^{-1}\\right)[\/latex]\r\n\r\n[latex]\\Large\\frac{6\\times {10}^{9}}{2\\times {10}^{-1}}[\/latex]\r\n\r\n30,000,000,000\r\n\r\n[latex]\\Large\\frac{9\\times {10}^{-3}}{1\\times {10}^{-6}}[\/latex]\r\n

                Introduction to Factoring Polynomials<\/h2>\r\nFind the Greatest Common Factor of Two or More Expressions<\/strong>\r\nIn the following exercises, find the greatest common factor.\r\n\r\n[latex]5n,45[\/latex]\r\n\r\n5\r\n\r\n[latex]8a,72[\/latex]\r\n\r\n[latex]12{x}^{2},20{x}^{3},36{x}^{4}[\/latex]\r\n\r\n4x<\/em>2<\/sup>\r\n\r\n[latex]9{y}^{4},21{y}^{5},15{y}^{6}[\/latex]\r\n\r\nFactor the Greatest Common Factor from a Polynomial<\/strong>\r\nIn the following exercises, factor the greatest common factor from each polynomial.\r\n\r\n[latex]16u - 24[\/latex]\r\n\r\n8(2u<\/em> \u2212 3)\r\n\r\n[latex]15r+35[\/latex]\r\n\r\n[latex]6{p}^{2}+6p[\/latex]\r\n\r\n6p<\/em>(p<\/em> + 1)\r\n\r\n[latex]10{c}^{2}-10c[\/latex]\r\n\r\n[latex]-9{a}^{5}-9{a}^{3}[\/latex]\r\n\r\n\u22129a<\/em>3<\/sup>(a<\/em>2<\/sup> + 1)\r\n\r\n[latex]-7{x}^{8}-28{x}^{3}[\/latex]\r\n\r\n[latex]5{y}^{2}-55y+45[\/latex]\r\n\r\n5(y<\/em>2<\/sup> \u2212 11y<\/em> + 9)\r\n\r\n[latex]2{q}^{5}-16{q}^{3}+30{q}^{2}[\/latex]\r\n

                Chapter Practice Test<\/h1>\r\nFor the polynomial [latex]8{y}^{4}-3{y}^{2}+1[\/latex]\r\n
                  \r\n \t
                1. \u24d0 Is it a monomial, binomial, or trinomial?<\/li>\r\n \t
                2. \u24d1 What is its degree?<\/li>\r\n<\/ol>\r\n
                    \r\n \t
                  1. \u24d0 trinomial<\/li>\r\n \t
                  2. \u24d1 4<\/li>\r\n<\/ol>\r\nIn the following exercises, simplify each expression.\r\n\r\n[latex]\\left(5{a}^{2}+2a - 12\\right)+\\left(9{a}^{2}+8a - 4\\right)[\/latex]\r\n\r\n[latex]\\left(10{x}^{2}-3x+5\\right)-\\left(4{x}^{2}-6\\right)[\/latex]\r\n\r\n6x<\/em>2<\/sup> \u2212 3x<\/em> + 11\r\n\r\n[latex]{\\left(-\\Large\\frac{3}{4}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[latex]n\\cdot {n}^{4}[\/latex]\r\n\r\nn<\/em>5<\/sup>\r\n\r\n[latex]{\\left(10{p}^{3}{q}^{5}\\right)}^{2}[\/latex]\r\n\r\n[latex]\\left(8x{y}^{3}\\right)\\left(-6{x}^{4}{y}^{6}\\right)[\/latex]\r\n\r\n\u221248x<\/em>5<\/sup>y<\/em>9<\/sup>\r\n\r\n[latex]4u\\left({u}^{2}-9u+1\\right)[\/latex]\r\n\r\n[latex]\\left(s+8\\right)\\left(s+9\\right)[\/latex]\r\n\r\ns<\/em>2<\/sup> + 17s<\/em> + 72\r\n\r\n[latex]\\left(m+3\\right)\\left(7m - 2\\right)[\/latex]\r\n\r\n[latex]\\left(11a - 6\\right)\\left(5a - 1\\right)[\/latex]\r\n\r\n55a<\/em>2<\/sup> \u2212 41a<\/em> + 6\r\n\r\n[latex]\\left(n - 8\\right)\\left({n}^{2}-4n+11\\right)[\/latex]\r\n\r\n[latex]\\left(4a+9b\\right)\\left(6a - 5b\\right)[\/latex]\r\n\r\n24a<\/em>2<\/sup> + 34ab<\/em> \u2212 45b<\/em>2<\/sup>\r\n\r\n[latex]\\Large\\frac{{5}^{6}}{{5}^{8}}[\/latex]\r\n\r\n[latex]{\\left(\\Large\\frac{{x}^{3}\\cdot {x}^{9}}{{x}^{5}}\\normalsize\\right)}^{2}[\/latex]\r\n\r\nx<\/em>14<\/sup>\r\n\r\n[latex]{\\left(47{a}^{18}{b}^{23}{c}^{5}\\right)}^{0}[\/latex]\r\n\r\n[latex]\\Large\\frac{24{r}^{3}s}{6{r}^{2}{s}^{7}}[\/latex]\r\n\r\n[latex]\\Large\\frac{4r}{{s}^{6}}[\/latex]\r\n\r\n[latex]\\Large\\frac{8{y}^{2}-16y+20}{4y}[\/latex]\r\n\r\n[latex]\\left(15x{y}^{3}-35{x}^{2}y\\right)\\div 5xy[\/latex]\r\n\r\n3y<\/em>2<\/sup> \u2212 7x<\/em>\r\n\r\n[latex]{4}^{-1}[\/latex]\r\n\r\n[latex]{\\left(2y\\right)}^{-3}[\/latex]\r\n\r\n[latex]\\Large\\frac{1}{2y}[\/latex]\r\n\r\n[latex]{p}^{-3}\\cdot {p}^{-8}[\/latex]\r\n\r\n[latex]\\Large\\frac{{x}^{4}}{{x}^{-5}}[\/latex]\r\n\r\nx<\/em>9<\/sup>\r\n\r\n[latex]\\left(2.4\\times {10}^{8}\\right)\\left(2\\times {10}^{-5}\\right)[\/latex]\r\n\r\nIn the following exercises, factor the greatest common factor from each polynomial.\r\n\r\n[latex]80{a}^{3}+120{a}^{2}+40a[\/latex]\r\n\r\n[latex]-6{x}^{2}-30x[\/latex]\r\n\r\n\u22126x<\/em>(x<\/em> + 5)\r\n\r\nConvert [latex]5.25\\times {10}^{-4}[\/latex] to decimal form.\r\n\r\n0.000525\r\n\r\nIn the following exercises, simplify, and write your answer in decimal form.\r\n\r\n[latex]\\Large\\frac{9\\times {10}^{4}}{3\\times {10}^{-1}}[\/latex]\r\n\r\n3 \u00d7 105<\/sup>\r\n\r\nA hiker drops a pebble from a bridge [latex]240[\/latex] feet above a canyon. The polynomial [latex]-16{t}^{2}+240[\/latex] gives the height of the pebble [latex]t[\/latex] seconds a after it was dropped. Find the height when [latex]t=3[\/latex].\r\n\r\nAccording to www.cleanair.org, the amount of trash generated in the US in one year averages out to [latex]112,000[\/latex] pounds of trash per person. Write this number in scientific notation.\r\n

                    <\/h2>\r\n

                    Contribute!<\/h2>
                    Did you have an idea for improving this content? We\u2019d love your input.<\/div>Improve this page<\/a>Learn More<\/a>","rendered":"

                    Practice Makes Perfect<\/h2>\n

                    Identify Polynomials, Monomials, Binomials and Trinomials<\/strong>
                    \nIn the following exercises, determine if each of the polynomials is a monomial, binomial, trinomial, or other polynomial.<\/p>\n

                    [latex]5x+2[\/latex]<\/p>\n

                    binomial<\/p>\n

                    [latex]{z}^{2}-5z - 6[\/latex]<\/p>\n

                    [latex]{a}^{2}+9a+18[\/latex]<\/p>\n

                    trinomial<\/p>\n

                    [latex]-12{p}^{4}[\/latex]<\/p>\n

                    [latex]{y}^{3}-8{y}^{2}+2y - 16[\/latex]<\/p>\n

                    polynomial<\/p>\n

                    [latex]10 - 9x[\/latex]<\/p>\n

                    [latex]23{y}^{2}[\/latex]<\/p>\n

                    monomial<\/p>\n

                    [latex]{m}^{4}+4{m}^{3}+6{m}^{2}+4m+1[\/latex]<\/p>\n

                    Determine the Degree of Polynomials<\/strong>
                    \nIn the following exercises, determine the degree of each polynomial.<\/p>\n

                    [latex]8{a}^{5}-2{a}^{3}+1[\/latex]<\/p>\n

                    5<\/p>\n

                    [latex]5{c}^{3}+11{c}^{2}-c - 8[\/latex]<\/p>\n

                    [latex]3x - 12[\/latex]<\/p>\n

                    1<\/p>\n

                    [latex]4y+17[\/latex]<\/p>\n

                    [latex]-13[\/latex]<\/p>\n

                    0<\/p>\n

                    [latex]-22[\/latex]<\/p>\n

                    Add and Subtract Monomials<\/strong>
                    \nIn the following exercises, add or subtract the monomials.<\/p>\n

                    [latex]{\\text{6x}}^{2}+9{x}^{2}[\/latex]<\/p>\n

                    15x<\/em>2<\/sup><\/p>\n

                    [latex]{\\text{4y}}^{3}+6{y}^{3}[\/latex]<\/p>\n

                    [latex]-12u+4u[\/latex]<\/p>\n

                    \u22128u<\/em><\/p>\n

                    [latex]-3m+9m[\/latex]<\/p>\n

                    [latex]5a+7b[\/latex]<\/p>\n

                    5a<\/em> + 7b<\/em><\/p>\n

                    [latex]8y+6z[\/latex]<\/p>\n

                    Add: [latex]\\text{}4a,-3b,-8a[\/latex]<\/p>\n

                    \u22124a<\/em> \u22123b<\/em><\/p>\n

                    Add: [latex]4x,3y,-3x[\/latex]<\/p>\n

                    [latex]18x - 2x[\/latex]<\/p>\n

                    16x<\/em><\/p>\n

                    [latex]13a - 3a[\/latex]<\/p>\n

                    Subtract [latex]5{x}^{6}\\text{from}-12{x}^{6}[\/latex]<\/p>\n

                    \u221217x<\/em>6<\/sup><\/p>\n

                    Subtract [latex]2{p}^{4}\\text{from}-7{p}^{4}[\/latex]<\/p>\n

                    Add and Subtract Polynomials<\/strong>
                    \nIn the following exercises, add or subtract the polynomials.<\/p>\n

                    [latex]\\left(4{y}^{2}+10y+3\\right)+\\left(8{y}^{2}-6y+5\\right)[\/latex]<\/p>\n

                    12y<\/em>2<\/sup> + 4y<\/em> + 8<\/p>\n

                    [latex]\\left(7{x}^{2}-9x+2\\right)+\\left(6{x}^{2}-4x+3\\right)[\/latex]<\/p>\n

                    [latex]\\left({x}^{2}+6x+8\\right)+\\left(-4{x}^{2}+11x - 9\\right)[\/latex]<\/p>\n

                    \u22123x<\/em>2<\/sup> + 17x<\/em> \u2212 1<\/p>\n

                    [latex]\\left({y}^{2}+9y+4\\right)+\\left(-2{y}^{2}-5y - 1\\right)[\/latex]<\/p>\n

                    [latex]\\left(3{a}^{2}+7\\right)+\\left({a}^{2}-7a - 18\\right)[\/latex]<\/p>\n

                    4a<\/em>2<\/sup> \u2212 7a<\/em> \u2212 11<\/p>\n

                    [latex]\\left({p}^{2}-5p - 11\\right)+\\left(3{p}^{2}+9\\right)[\/latex]<\/p>\n

                    [latex]\\left(6{m}^{2}-9m - 3\\right)-\\left(2{m}^{2}+m - 5\\right)[\/latex]<\/p>\n

                    4m<\/em>2<\/sup> \u2212 10m<\/em> + 2<\/p>\n

                    [latex]\\left(3{n}^{2}-4n+1\\right)-\\left(4{n}^{2}-n - 2\\right)[\/latex]<\/p>\n

                    [latex]\\left({z}^{2}+8z+9\\right)-\\left({z}^{2}-3z+1\\right)[\/latex]<\/p>\n

                    11z<\/em> + 8<\/p>\n

                    [latex]\\left({z}^{2}-7z+5\\right)-\\left({z}^{2}-8z+6\\right)[\/latex]<\/p>\n

                    [latex]\\left(12{s}^{2}-15s\\right)-\\left(s - 9\\right)[\/latex]<\/p>\n

                    12s<\/em>2<\/sup> \u2212 16s<\/em> + 9<\/p>\n

                    [latex]\\left(10{r}^{2}-20r\\right)-\\left(r - 8\\right)[\/latex]<\/p>\n

                    Find the sum of [latex]\\left(2{p}^{3}-8\\right)[\/latex] and [latex]\\left({p}^{2}+9p+18\\right)[\/latex]<\/p>\n

                    2p<\/em>3<\/sup> + p<\/em>2<\/sup> + 9p<\/em> + 10<\/p>\n

                    Find the sum of [latex]\\left({q}^{2}+4q+13\\right)[\/latex] and [latex]\\left(7{q}^{3}-3\\right)[\/latex]<\/p>\n

                    Subtract [latex]\\left(7{x}^{2}-4x+2\\right)[\/latex] from [latex]\\left(8{x}^{2}-x+6\\right)[\/latex]<\/p>\n

                    x<\/em>2<\/sup> + 3x<\/em> + 4<\/p>\n

                    Subtract [latex]\\left(5{x}^{2}-x+12\\right)[\/latex] from [latex]\\left(9{x}^{2}-6x - 20\\right)[\/latex]<\/p>\n

                    Find the difference of [latex]\\left({w}^{2}+w - 42\\right)[\/latex] and [latex]\\left({w}^{2}-10w+24\\right)[\/latex]<\/p>\n

                    11w<\/em> \u2212 66<\/p>\n

                    Find the difference of [latex]\\left({z}^{2}-3z - 18\\right)[\/latex] and [latex]\\left({z}^{2}+5z - 20\\right)[\/latex]<\/p>\n

                    Evaluate a Polynomial for a Given Value<\/strong>
                    \nIn the following exercises, evaluate each polynomial for the given value.<\/p>\n

                    [latex]\\text{Evaluate}8{y}^{2}-3y+2[\/latex]<\/p>\n

                    \u24d0 [latex]y=5[\/latex]
                    \n\u24d1 [latex]y=-2[\/latex]
                    \n\u24d2 [latex]y=0[\/latex]<\/p>\n

                    \u24d0 187
                    \n\u24d1 40
                    \n\u24d2 2<\/p>\n

                    [latex]\\text{Evaluate}5{y}^{2}-y - 7\\text{when:}[\/latex]<\/p>\n

                    \u24d0 [latex]y=-4[\/latex]
                    \n\u24d1 [latex]y=1[\/latex]
                    \n\u24d2 [latex]y=0[\/latex]<\/p>\n

                    [latex]\\text{Evaluate}4 - 36x\\text{when:}[\/latex]<\/p>\n

                    \u24d0 [latex]x=3[\/latex]
                    \n\u24d1 [latex]x=0[\/latex]
                    \n\u24d2 [latex]x=-1[\/latex]<\/p>\n

                    \u24d0 \u2212104
                    \n\u24d1 4
                    \n\u24d2 40<\/p>\n

                    [latex]\\text{Evaluate}16 - 36{x}^{2}\\text{when:}[\/latex]<\/p>\n

                    \u24d0 [latex]x=-1[\/latex]
                    \n\u24d1 [latex]x=0[\/latex]
                    \n\u24d2 [latex]x=2[\/latex]<\/p>\n

                    A window washer drops a squeegee from a platform [latex]275[\/latex] feet high. The polynomial [latex]-16{t}^{2}+275[\/latex] gives the height of the squeegee [latex]t[\/latex] seconds after it was dropped. Find the height after [latex]t=4[\/latex] seconds.<\/p>\n

                    19 feet<\/p>\n

                    A manufacturer of microwave ovens has found that the revenue received from selling microwaves at a cost of p<\/em> dollars each is given by the polynomial [latex]-5{p}^{2}+350p[\/latex]. Find the revenue received when [latex]p=50[\/latex] dollars.<\/p>\n

                    Everyday Math<\/h2>\n

                    Fuel Efficiency<\/strong> The fuel efficiency (in miles per gallon) of a bus going at a speed of [latex]x[\/latex] miles per hour is given by the polynomial [latex]-\\Large\\frac{1}{160}\\normalsize{x}^{2}+\\Large\\frac{1}{2}\\normalsize x[\/latex]. Find the fuel efficiency when [latex]x=40\\text{mph.}[\/latex]<\/p>\n

                    10 mpg<\/p>\n

                    Stopping Distance<\/strong> The number of feet it takes for a car traveling at [latex]x[\/latex] miles per hour to stop on dry, level concrete is given by the polynomial [latex]0.06{x}^{2}+1.1x[\/latex]. Find the stopping distance when [latex]x=60\\text{mph.}[\/latex]<\/p>\n

                    Writing Exercises<\/h2>\n

                    Using your own words, explain the difference between a monomial, a binomial, and a trinomial.<\/p>\n

                    Answers will vary.<\/p>\n

                    Eloise thinks the sum [latex]5{x}^{2}+3{x}^{4}[\/latex] is [latex]8{x}^{6}[\/latex]. What is wrong with her reasoning?<\/p>\n

                    Self Check<\/h2>\n

                    \u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n

                    \".\"
                    \n\u24d1 If most of your checks were:
                    \n\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
                    \n\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
                    \n\u2026no\u2014I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n

                    <\/h2>\n

                    Practice Makes Perfect<\/h2>\n

                    Simplify Expressions with Exponents<\/strong>
                    \nIn the following exercises, simplify each expression with exponents.<\/p>\n

                    [latex]{4}^{5}[\/latex]<\/p>\n

                    1,024<\/p>\n

                    [latex]{10}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                    [latex]\\Large\\frac{1}{4}[\/latex]<\/p>\n

                    [latex]{\\left(\\Large\\frac{3}{5}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left(0.2\\right)}^{3}[\/latex]<\/p>\n

                    0.008<\/p>\n

                    [latex]{\\left(0.4\\right)}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(-5\\right)}^{4}[\/latex]<\/p>\n

                    625<\/p>\n

                    [latex]{\\left(-3\\right)}^{5}[\/latex]<\/p>\n

                    [latex]{-5}^{4}[\/latex]<\/p>\n

                    \u2212625<\/p>\n

                    [latex]{-3}^{5}[\/latex]<\/p>\n

                    [latex]{-10}^{4}[\/latex]<\/p>\n

                    \u221210,000<\/p>\n

                    [latex]{-2}^{6}[\/latex]<\/p>\n

                    [latex]{\\left(-\\Large\\frac{2}{3}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                    [latex]-\\Large\\frac{8}{27}[\/latex]<\/p>\n

                    [latex]{\\left(-\\Large\\frac{1}{4}\\normalsize\\right)}^{4}[\/latex]<\/p>\n

                    [latex]-{0.5}^{2}[\/latex]<\/p>\n

                    \u2212.25<\/p>\n

                    [latex]-{0.1}^{4}[\/latex]<\/p>\n

                    Simplify Expressions Using the Product Property of Exponents<\/strong>
                    \nIn the following exercises, simplify each expression using the Product Property of Exponents.<\/p>\n

                    [latex]{x}^{3}\\cdot {x}^{6}[\/latex]<\/p>\n

                    x<\/em>9<\/sup><\/p>\n

                    [latex]{m}^{4}\\cdot {m}^{2}[\/latex]<\/p>\n

                    [latex]a\\cdot {a}^{4}[\/latex]<\/p>\n

                    a<\/em>5<\/sup><\/p>\n

                    [latex]{y}^{12}\\cdot y[\/latex]<\/p>\n

                    [latex]{3}^{5}\\cdot {3}^{9}[\/latex]<\/p>\n

                    314<\/sup><\/p>\n

                    [latex]{5}^{10}\\cdot {5}^{6}[\/latex]<\/p>\n

                    [latex]z\\cdot {z}^{2}\\cdot {z}^{3}[\/latex]<\/p>\n

                    z<\/em>6<\/sup><\/p>\n

                    [latex]a\\cdot {a}^{3}\\cdot {a}^{5}[\/latex]<\/p>\n

                    [latex]{x}^{a}\\cdot {x}^{2}[\/latex]<\/p>\n

                    x<\/em>a<\/em>+2<\/sup><\/p>\n

                    [latex]{y}^{p}\\cdot {y}^{3}[\/latex]<\/p>\n

                    [latex]{y}^{a}\\cdot {y}^{b}[\/latex]<\/p>\n

                    y<\/em>a<\/em>+b<\/em><\/sup><\/p>\n

                    [latex]{x}^{p}\\cdot {x}^{q}[\/latex]<\/p>\n

                    Simplify Expressions Using the Power Property of Exponents<\/strong>
                    \nIn the following exercises, simplify each expression using the Power Property of Exponents.<\/em><\/p>\n

                    [latex]{\\left({u}^{4}\\right)}^{2}[\/latex]<\/p>\n

                    u<\/em>8<\/sup><\/p>\n

                    [latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/p>\n

                    [latex]{\\left({y}^{5}\\right)}^{4}[\/latex]<\/p>\n

                    y<\/em>20<\/sup><\/p>\n

                    [latex]{\\left({a}^{3}\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left({10}^{2}\\right)}^{6}[\/latex]<\/p>\n

                    1012<\/sup><\/p>\n

                    [latex]{\\left({2}^{8}\\right)}^{3}[\/latex]<\/p>\n

                    [latex]{\\left({x}^{15}\\right)}^{6}[\/latex]<\/p>\n

                    x<\/em>90<\/sup><\/p>\n

                    [latex]{\\left({y}^{12}\\right)}^{8}[\/latex]<\/p>\n

                    [latex]{\\left({x}^{2}\\right)}^{y}[\/latex]<\/p>\n

                    x<\/em>2y<\/em><\/sup><\/p>\n

                    [latex]{\\left({y}^{3}\\right)}^{x}[\/latex]<\/p>\n

                    [latex]{\\left({5}^{x}\\right)}^{y}[\/latex]<\/p>\n

                    5x<\/em>y<\/em><\/sup><\/p>\n

                    [latex]{\\left({7}^{a}\\right)}^{b}[\/latex]<\/p>\n

                    Simplify Expressions Using the Product to a Power Property<\/strong>
                    \nIn the following exercises, simplify each expression using the Product to a Power Property.<\/p>\n

                    [latex]{\\left(5a\\right)}^{2}[\/latex]<\/p>\n

                    25a<\/em>2<\/sup><\/p>\n

                    [latex]{\\left(7x\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left(-6m\\right)}^{3}[\/latex]<\/p>\n

                    \u2212216m<\/em>3<\/sup><\/p>\n

                    [latex]{\\left(-9n\\right)}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(4rs\\right)}^{2}[\/latex]<\/p>\n

                    16r<\/em>2<\/sup>s<\/em>2<\/sup><\/p>\n

                    [latex]{\\left(5ab\\right)}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(4xyz\\right)}^{4}[\/latex]<\/p>\n

                    256x<\/em>4<\/sup>y<\/em>4<\/sup>z<\/em>4<\/sup><\/p>\n

                    [latex]{\\left(-5abc\\right)}^{3}[\/latex]<\/p>\n

                    Simplify Expressions by Applying Several Properties<\/strong>
                    \nIn the following exercises, simplify each expression.<\/p>\n

                    [latex]{\\left({x}^{2}\\right)}^{4}\\cdot {\\left({x}^{3}\\right)}^{2}[\/latex]<\/p>\n

                    x<\/em>14<\/sup><\/p>\n

                    [latex]{\\left({y}^{4}\\right)}^{3}\\cdot {\\left({y}^{5}\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left({a}^{2}\\right)}^{6}\\cdot {\\left({a}^{3}\\right)}^{8}[\/latex]<\/p>\n

                    a<\/em>36<\/sup><\/p>\n

                    [latex]{\\left({b}^{7}\\right)}^{5}\\cdot {\\left({b}^{2}\\right)}^{6}[\/latex]<\/p>\n

                    [latex]{\\left(3x\\right)}^{2}\\left(5x\\right)[\/latex]<\/p>\n

                    45x<\/em>3<\/sup><\/p>\n

                    [latex]{\\left(2y\\right)}^{3}\\left(6y\\right)[\/latex]<\/p>\n

                    [latex]{\\left(5a\\right)}^{2}{\\left(2a\\right)}^{3}[\/latex]<\/p>\n

                    200a<\/em>5<\/sup><\/p>\n

                    [latex]{\\left(4b\\right)}^{2}{\\left(3b\\right)}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(2{m}^{6}\\right)}^{3}[\/latex]<\/p>\n

                    8m<\/em>18<\/sup><\/p>\n

                    [latex]{\\left(3{y}^{2}\\right)}^{4}[\/latex]<\/p>\n

                    [latex]{\\left(10{x}^{2}y\\right)}^{3}[\/latex]<\/p>\n

                    1,000x<\/em>6<\/sup>y<\/em>3<\/sup><\/p>\n

                    [latex]{\\left(2m{n}^{4}\\right)}^{5}[\/latex]<\/p>\n

                    [latex]{\\left(-2{a}^{3}{b}^{2}\\right)}^{4}[\/latex]<\/p>\n

                    16a<\/em>12<\/sup>b<\/em>8<\/sup><\/p>\n

                    [latex]{\\left(-10{u}^{2}{v}^{4}\\right)}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(\\Large\\frac{2}{3}\\normalsize{x}^{2}y\\right)}^{3}[\/latex]<\/p>\n

                    [latex]\\Large\\frac{8}{27}\\normalsize{x}^{6}{y}^{3}[\/latex]<\/p>\n

                    [latex]{\\left(\\Large\\frac{7}{9}\\normalsize p{q}^{4}\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left(8{a}^{3}\\right)}^{2}{\\left(2a\\right)}^{4}[\/latex]<\/p>\n

                    1,024a<\/em>10<\/sup><\/p>\n

                    [latex]{\\left(5{r}^{2}\\right)}^{3}{\\left(3r\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left(10{p}^{4}\\right)}^{3}{\\left(5{p}^{6}\\right)}^{2}[\/latex]<\/p>\n

                    25,000p<\/em>24<\/sup><\/p>\n

                    [latex]{\\left(4{x}^{3}\\right)}^{3}{\\left(2{x}^{5}\\right)}^{4}[\/latex]<\/p>\n

                    [latex]{\\left(\\Large\\frac{1}{2}\\normalsize{x}^{2}{y}^{3}\\right)}^{4}{\\left(4{x}^{5}{y}^{3}\\right)}^{2}[\/latex]<\/p>\n

                    x<\/em>18<\/sup>y<\/em>18<\/sup><\/p>\n

                    [latex]{\\left(\\Large\\frac{1}{3}\\normalsize{m}^{3}{n}^{2}\\right)}^{4}{\\left(9{m}^{8}{n}^{3}\\right)}^{2}[\/latex]<\/p>\n

                    [latex]{\\left(3{m}^{2}n\\right)}^{2}{\\left(2m{n}^{5}\\right)}^{4}[\/latex]<\/p>\n

                    144m<\/em>8<\/sup>n<\/em>22<\/sup><\/p>\n

                    [latex]{\\left(2p{q}^{4}\\right)}^{3}{\\left(5{p}^{6}q\\right)}^{2}[\/latex]<\/p>\n

                    Multiply Monomials<\/strong>
                    \nIn the following exercises, multiply the following monomials.<\/p>\n

                    [latex]\\left(12{x}^{2}\\right)\\left(-5{x}^{4}\\right)[\/latex]<\/p>\n

                    \u221260x<\/em>6<\/sup><\/p>\n

                    [latex]\\left(-10{y}^{3}\\right)\\left(7{y}^{2}\\right)[\/latex]<\/p>\n

                    [latex]\\left(-8{u}^{6}\\right)\\left(-9u\\right)[\/latex]<\/p>\n

                    72u<\/em>7<\/sup><\/p>\n

                    [latex]\\left(-6{c}^{4}\\right)\\left(-12c\\right)[\/latex]<\/p>\n

                    [latex]\\left(\\Large\\frac{1}{5}\\normalsize{r}^{8}\\right)\\left(20{r}^{3}\\right)[\/latex]<\/p>\n

                    4r<\/em>11<\/sup><\/p>\n

                    [latex]\\left(\\Large\\frac{1}{4}\\normalsize{a}^{5}\\right)\\left(36{a}^{2}\\right)[\/latex]<\/p>\n

                    [latex]\\left(4{a}^{3}b\\right)\\left(9{a}^{2}{b}^{6}\\right)[\/latex]<\/p>\n

                    36a<\/em>5<\/sup>b<\/em>7<\/sup><\/p>\n

                    [latex]\\left(6{m}^{4}{n}^{3}\\right)\\left(7m{n}^{5}\\right)[\/latex]<\/p>\n

                    [latex]\\left(\\Large\\frac{4}{7}\\normalsize x{y}^{2}\\right)\\left(14x{y}^{3}\\right)[\/latex]<\/p>\n

                    8x<\/em>2<\/sup>y<\/em>5<\/sup><\/p>\n

                    [latex]\\left(\\Large\\frac{5}{8}\\normalsize{u}^{3}v\\right)\\left(24{u}^{5}v\\right)[\/latex]<\/p>\n

                    [latex]\\left(\\Large\\frac{2}{3}\\normalsize{x}^{2}y\\right)\\left(\\Large\\frac{3}{4}\\normalsize x{y}^{2}\\right)[\/latex]<\/p>\n

                    [latex]\\Large\\frac{1}{2}\\normalsize{x}^{3}{y}^{3}[\/latex]<\/p>\n

                    [latex]\\left(\\Large\\frac{3}{5}\\normalsize{m}^{3}{n}^{2}\\right)\\left(\\Large\\frac{5}{9}\\normalsize{m}^{2}{n}^{3}\\right)[\/latex]<\/p>\n

                    Everyday Math<\/h2>\n

                    Email<\/strong> Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is [latex]{6}^{2}[\/latex], on the third round is [latex]{6}^{3}[\/latex], as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.<\/p>\n\n\n\n\n\n\n\n\n\n
                    Round<\/th>\nNumber of people<\/th>\n<\/tr>\n<\/thead>\n
                    [latex]1[\/latex]<\/td>\n[latex]6[\/latex]<\/td>\n<\/tr>\n
                    [latex]2[\/latex]<\/td>\n[latex]{6}^{2}[\/latex]<\/td>\n<\/tr>\n
                    [latex]3[\/latex]<\/td>\n[latex]{6}^{3}[\/latex]<\/td>\n<\/tr>\n
                    [latex]\\dots[\/latex]<\/td>\n[latex]\\dots[\/latex]<\/td>\n<\/tr>\n
                    [latex]8[\/latex]<\/td>\n[latex]?[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                    1,679,616<\/p>\n

                    Salary<\/strong> Raul\u2019s boss gives him a [latex]\\text{5%}[\/latex] raise every year on his birthday. This means that each year, Raul\u2019s salary is [latex]1.05[\/latex] times his last year\u2019s salary. If his original salary was [latex]{$40,000}[\/latex] , his salary after [latex]1[\/latex] year was [latex]{$40,000}\\left(1.05\\right)[\/latex], after [latex]2[\/latex] years was [latex]{$40,000}{\\left(1.05\\right)}^{2}[\/latex], after [latex]3[\/latex] years was [latex]{$40,000}{\\left(1.05\\right)}^{3}[\/latex], as shown in the table below. What will Raul\u2019s salary be after [latex]10[\/latex] years? Simplify the expression, to show Raul\u2019s salary in dollars.<\/p>\n\n\n\n\n\n\n\n\n\n
                    Year<\/th>\nSalary<\/th>\n<\/tr>\n<\/thead>\n
                    [latex]1[\/latex]<\/td>\n[latex]{$40,000}\\left(1.05\\right)[\/latex]<\/td>\n<\/tr>\n
                    [latex]2[\/latex]<\/td>\n[latex]{$40,000}{\\left(1.05\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n
                    [latex]3[\/latex]<\/td>\n[latex]{$40,000}{\\left(1.05\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n
                    [latex]\\dots[\/latex]<\/td>\n[latex]\\dots[\/latex]<\/td>\n<\/tr>\n
                    [latex]10[\/latex]<\/td>\n[latex]?[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                    Writing Exercises<\/h2>\n

                    Use the Product Property for Exponents to explain why [latex]x\\cdot x={x}^{2}[\/latex].<\/p>\n

                    Answers will vary.<\/p>\n

                    Explain why [latex]{-5}^{3}={\\left(-5\\right)}^{3}[\/latex] but [latex]{-5}^{4}\\ne {\\left(-5\\right)}^{4}[\/latex].<\/p>\n

                    Jorge thinks [latex]{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{2}[\/latex] is [latex]1[\/latex]. What is wrong with his reasoning?<\/p>\n

                    Answers will vary.<\/p>\n

                    Explain why [latex]{x}^{3}\\cdot {x}^{5}[\/latex] is [latex]{x}^{8}[\/latex], and not [latex]{x}^{15}[\/latex].<\/p>\n

                    Self Check<\/h2>\n

                    \u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n

                    \".\"
                    \n\u24d1 After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n

                     <\/p>\n

                    Practice Makes Perfect<\/h2>\n

                    Multiply a Polynomial by a Monomial<\/strong>
                    \nIn the following exercises, multiply.<\/p>\n

                    [latex]4\\left(x+10\\right)[\/latex]<\/p>\n

                    4x<\/em> + 40<\/p>\n

                    [latex]6\\left(y+8\\right)[\/latex]<\/p>\n

                    [latex]15\\left(r - 24\\right)[\/latex]<\/p>\n

                    15r<\/em> \u2212 360<\/p>\n

                    [latex]12\\left(v - 30\\right)[\/latex]<\/p>\n

                    [latex]-3\\left(m+11\\right)[\/latex]<\/p>\n

                    \u22123m<\/em> \u2212 33<\/p>\n

                    [latex]-4\\left(p+15\\right)[\/latex]<\/p>\n

                    [latex]-8\\left(z - 5\\right)[\/latex]<\/p>\n

                    \u22128z<\/em> + 40<\/p>\n

                    [latex]-3\\left(x - 9\\right)[\/latex]<\/p>\n

                    [latex]u\\left(u+5\\right)[\/latex]<\/p>\n

                    u<\/em>2<\/sup> + 5u<\/em><\/p>\n

                    [latex]q\\left(q+7\\right)[\/latex]<\/p>\n

                    [latex]n\\left({n}^{2}-3n\\right)[\/latex]<\/p>\n

                    n<\/em>3<\/sup> \u2212 3n<\/em>2<\/sup><\/p>\n

                    [latex]s\\left({s}^{2}-6s\\right)[\/latex]<\/p>\n

                    [latex]12x\\left(x - 10\\right)[\/latex]<\/p>\n

                    12x<\/em>2<\/sup> \u2212 120x<\/em><\/p>\n

                    [latex]9m\\left(m - 11\\right)[\/latex]<\/p>\n

                    [latex]-9a\\left(3a+5\\right)[\/latex]<\/p>\n

                    \u221227a<\/em>2<\/sup> \u2212 45a<\/em><\/p>\n

                    [latex]-4p\\left(2p+7\\right)[\/latex]<\/p>\n

                    [latex]6x\\left(4x+y\\right)[\/latex]<\/p>\n

                    24x<\/em>2<\/sup> + 6xy<\/em><\/p>\n

                    [latex]5a\\left(9a+b\\right)[\/latex]<\/p>\n

                    [latex]5p\\left(11p - 5q\\right)[\/latex]<\/p>\n

                    55p<\/em>2<\/sup> \u2212 25pq<\/em><\/p>\n

                    [latex]12u\\left(3u - 4v\\right)[\/latex]<\/p>\n

                    [latex]3\\left({v}^{2}+10v+25\\right)[\/latex]<\/p>\n

                    3v<\/em>2<\/sup> + 30v<\/em> + 75<\/p>\n

                    [latex]6\\left({x}^{2}+8x+16\\right)[\/latex]<\/p>\n

                    [latex]2n\\left(4{n}^{2}-4n+1\\right)[\/latex]<\/p>\n

                    8n<\/em>3<\/sup> \u2212 8n<\/em>2<\/sup> + 2n<\/em><\/p>\n

                    [latex]3r\\left(2{r}^{2}-6r+2\\right)[\/latex]<\/p>\n

                    [latex]-8y\\left({y}^{2}+2y - 15\\right)[\/latex]<\/p>\n

                    \u22128y<\/em>3<\/sup> \u2212 16y<\/em>2<\/sup> + 120y<\/em><\/p>\n

                    [latex]-5m\\left({m}^{2}+3m - 18\\right)[\/latex]<\/p>\n

                    [latex]5{q}^{3}\\left({q}^{2}-2q+6\\right)[\/latex]<\/p>\n

                    5q<\/em>5<\/sup> \u2212 10q<\/em>4<\/sup> + 30q<\/em>3<\/sup><\/p>\n

                    [latex]9{r}^{3}\\left({r}^{2}-3r+5\\right)[\/latex]<\/p>\n

                    [latex]-4{z}^{2}\\left(3{z}^{2}+12z - 1\\right)[\/latex]<\/p>\n

                    \u221212z<\/em>4<\/sup> \u2212 48z<\/em>3<\/sup> + 4z<\/em>2<\/sup><\/p>\n

                    [latex]-3{x}^{2}\\left(7{x}^{2}+10x - 1\\right)[\/latex]<\/p>\n

                    [latex]\\left(2y - 9\\right)y[\/latex]<\/p>\n

                    2y<\/em>2<\/sup> \u2212 9y<\/em><\/p>\n

                    [latex]\\left(8b - 1\\right)b[\/latex]<\/p>\n

                    [latex]\\left(w - 6\\right)\\cdot 8[\/latex]<\/p>\n

                    8w<\/em> \u2212 48<\/p>\n

                    [latex]\\left(k - 4\\right)\\cdot 5[\/latex]<\/p>\n

                    Multiply a Binomial by a Binomial<\/strong>
                    \nIn the following exercises, multiply the following binomials using: \u24d0 the Distributive Property \u24d1 the FOIL method \u24d2 the Vertical method<\/p>\n

                    [latex]\\left(x+4\\right)\\left(x+6\\right)[\/latex]<\/p>\n

                    x<\/em>2<\/sup> + 10x<\/em> + 24<\/p>\n

                    [latex]\\left(u+8\\right)\\left(u+2\\right)[\/latex]<\/p>\n

                    [latex]\\left(n+12\\right)\\left(n - 3\\right)[\/latex]<\/p>\n

                    n<\/em>2<\/sup> + 9n<\/em> \u2212 36<\/p>\n

                    [latex]\\left(y+3\\right)\\left(y - 9\\right)[\/latex]<\/p>\n

                    In the following exercises, multiply the following binomials. Use any method.<\/p>\n

                    [latex]\\left(y+8\\right)\\left(y+3\\right)[\/latex]<\/p>\n

                    y<\/em>2<\/sup> + 11y<\/em> + 24<\/p>\n

                    [latex]\\left(x+5\\right)\\left(x+9\\right)[\/latex]<\/p>\n

                    [latex]\\left(a+6\\right)\\left(a+16\\right)[\/latex]<\/p>\n

                    a<\/em>2<\/sup> + 22a<\/em> + 96<\/p>\n

                    [latex]\\left(q+8\\right)\\left(q+12\\right)[\/latex]<\/p>\n

                    [latex]\\left(u - 5\\right)\\left(u - 9\\right)[\/latex]<\/p>\n

                    u<\/em>2<\/sup> \u2212 14u<\/em> + 45<\/p>\n

                    [latex]\\left(r - 6\\right)\\left(r - 2\\right)[\/latex]<\/p>\n

                    [latex]\\left(z - 10\\right)\\left(z - 22\\right)[\/latex]<\/p>\n

                    z<\/em>2<\/sup> \u2212 32z<\/em> + 220<\/p>\n

                    [latex]\\left(b - 5\\right)\\left(b - 24\\right)[\/latex]<\/p>\n

                    [latex]\\left(x - 4\\right)\\left(x+7\\right)[\/latex]<\/p>\n

                    x<\/em>2<\/sup> + 3x<\/em> \u2212 28<\/p>\n

                    [latex]\\left(s - 3\\right)\\left(s+8\\right)[\/latex]<\/p>\n

                    [latex]\\left(v+12\\right)\\left(v - 5\\right)[\/latex]<\/p>\n

                    v<\/em>2<\/sup> + 7v<\/em> \u2212 60<\/p>\n

                    [latex]\\left(d+15\\right)\\left(d - 4\\right)[\/latex]<\/p>\n

                    [latex]\\left(6n+5\\right)\\left(n+1\\right)[\/latex]<\/p>\n

                    6n<\/em>2<\/sup> + 11n<\/em> + 5<\/p>\n

                    [latex]\\left(7y+1\\right)\\left(y+3\\right)[\/latex]<\/p>\n

                    [latex]\\left(2m - 9\\right)\\left(10m+1\\right)[\/latex]<\/p>\n

                    20m<\/em>2<\/sup> \u2212 88m<\/em> \u2212 9<\/p>\n

                    [latex]\\left(5r - 4\\right)\\left(12r+1\\right)[\/latex]<\/p>\n

                    [latex]\\left(4c - 1\\right)\\left(4c+1\\right)[\/latex]<\/p>\n

                    16c<\/em>2<\/sup> \u2212 1<\/p>\n

                    [latex]\\left(8n - 1\\right)\\left(8n+1\\right)[\/latex]<\/p>\n

                    [latex]\\left(3u - 8\\right)\\left(5u - 14\\right)[\/latex]<\/p>\n

                    15u<\/em>2<\/sup> \u2212 82u<\/em> + 112<\/p>\n

                    [latex]\\left(2q - 5\\right)\\left(7q - 11\\right)[\/latex]<\/p>\n

                    [latex]\\left(a+b\\right)\\left(2a+3b\\right)[\/latex]<\/p>\n

                    2a<\/em>2<\/sup> + 5ab<\/em> + 3b<\/em>2<\/sup><\/p>\n

                    [latex]\\left(r+s\\right)\\left(3r+2s\\right)[\/latex]<\/p>\n

                    [latex]\\left(5x-y\\right)\\left(x - 4\\right)[\/latex]<\/p>\n

                    5x<\/em>2<\/sup> \u2212 20x<\/em> \u2212 xy<\/em> + 4y<\/em><\/p>\n

                    [latex]\\left(4z-y\\right)\\left(z - 6\\right)[\/latex]<\/p>\n

                    Multiply a Trinomial by a Binomial<\/strong>
                    \nIn the following exercises, multiply using \u24d0 the Distributive Property and \u24d1 the Vertical Method.<\/p>\n

                    [latex]\\left(u+4\\right)\\left({u}^{2}+3u+2\\right)[\/latex]<\/p>\n

                    u<\/em>3<\/sup> + 7u<\/em>2<\/sup> + 14u<\/em> + 8<\/p>\n

                    [latex]\\left(x+5\\right)\\left({x}^{2}+8x+3\\right)[\/latex]<\/p>\n

                    [latex]\\left(a+10\\right)\\left(3{a}^{2}+a - 5\\right)[\/latex]<\/p>\n

                    3a<\/em>3<\/sup> + 31a<\/em>2<\/sup> + 5a<\/em> \u2212 50<\/p>\n

                    [latex]\\left(n+8\\right)\\left(4{n}^{2}+n - 7\\right)[\/latex]<\/p>\n

                    In the following exercises, multiply. Use either method.<\/p>\n

                    [latex]\\left(y - 6\\right)\\left({y}^{2}-10y+9\\right)[\/latex]<\/p>\n

                    y<\/em>3<\/sup> \u2212 16y<\/em>2<\/sup> + 69y<\/em> \u2212 54<\/p>\n

                    [latex]\\left(k - 3\\right)\\left({k}^{2}-8k+7\\right)[\/latex]<\/p>\n

                    [latex]\\left(2x+1\\right)\\left({x}^{2}-5x - 6\\right)[\/latex]<\/p>\n

                    2x<\/em>3<\/sup> \u2212 9x<\/em>2<\/sup> \u2212 17x<\/em> \u2212 6<\/p>\n

                    [latex]\\left(5v+1\\right)\\left({v}^{2}-6v - 10\\right)[\/latex]<\/p>\n

                    Everyday Math<\/h2>\n

                    Mental math<\/strong> You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply [latex]13[\/latex] times [latex]15[\/latex]. Think of [latex]13[\/latex] as [latex]10+3[\/latex] and [latex]15[\/latex] as [latex]10+5[\/latex].<\/p>\n

                      \n
                    1. \u24d0 Multiply [latex]\\left(10+3\\right)\\left(10+5\\right)[\/latex] by the FOIL method.<\/li>\n
                    2. \u24d1 Multiply [latex]13\\cdot 15[\/latex] without using a calculator.<\/li>\n
                    3. \u24d2 Which way is easier for you? Why?<\/li>\n<\/ol>\n
                        \n
                      1. \u24d0 195<\/li>\n
                      2. \u24d1 195<\/li>\n
                      3. \u24d0 Answers will vary.<\/li>\n<\/ol>\n

                        Mental math<\/strong> You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply [latex]18[\/latex] times [latex]17[\/latex]. Think of [latex]18[\/latex] as [latex]20 - 2[\/latex] and [latex]17[\/latex] as [latex]20 - 3[\/latex].<\/p>\n

                          \n
                        1. \u24d0 Multiply [latex]\\left(20 - 2\\right)\\left(20 - 3\\right)[\/latex] by the FOIL method.<\/li>\n
                        2. \u24d1 Multiply [latex]18\\cdot 17[\/latex] without using a calculator.<\/li>\n
                        3. \u24d2 Which way is easier for you? Why?<\/li>\n<\/ol>\n

                          Writing Exercises<\/h2>\n

                          Which method do you prefer to use when multiplying two binomials\u2014the Distributive Property, the FOIL method, or the Vertical Method? Why?<\/p>\n

                          Answers will vary.<\/p>\n

                          Which method do you prefer to use when multiplying a trinomial by a binomial\u2014the Distributive Property or the Vertical Method? Why?<\/p>\n

                          Self Check<\/h2>\n

                          \u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n

                          \".\"
                          \n\u24d1 What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n

                          <\/h2>\n

                          Practice Makes Perfect<\/h2>\n

                          Simplify Expressions Using the Quotient Property of Exponents<\/strong>
                          \nIn the following exercises, simplify.<\/p>\n

                          [latex]\\Large\\frac{{4}^{8}}{{4}^{2}}[\/latex]<\/p>\n

                          46<\/sup><\/p>\n

                          [latex]\\Large\\frac{{3}^{12}}{{3}^{4}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{x}^{12}}{{x}^{3}}[\/latex]<\/p>\n

                          x<\/em>9<\/sup><\/p>\n

                          [latex]\\Large\\frac{{u}^{9}}{{u}^{3}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{r}^{5}}{r}[\/latex]<\/p>\n

                          r<\/em>4<\/sup><\/p>\n

                          [latex]\\Large\\frac{{y}^{4}}{y}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{y}^{4}}{{y}^{20}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{y}^{16}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{x}^{10}}{{x}^{30}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{10}^{3}}{{10}^{15}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{10}^{12}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{r}^{2}}{{r}^{8}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{a}{{a}^{9}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{a}^{8}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{2}{{2}^{5}}[\/latex]<\/p>\n

                          Simplify Expressions with Zero Exponents<\/strong>
                          \nIn the following exercises, simplify.<\/p>\n

                          [latex]{5}^{0}[\/latex]<\/p>\n

                          1<\/p>\n

                          [latex]{10}^{0}[\/latex]<\/p>\n

                          [latex]{a}^{0}[\/latex]<\/p>\n

                          1<\/p>\n

                          [latex]{x}^{0}[\/latex]<\/p>\n

                          [latex]-{7}^{0}[\/latex]<\/p>\n

                          \u22121<\/p>\n

                          [latex]-{4}^{0}[\/latex]<\/p>\n

                          \u24d0 [latex]{\\left(10p\\right)}^{0}[\/latex]
                          \n\u24d1 [latex]10{p}^{0}[\/latex]<\/p>\n

                          \u24d0 1
                          \n\u24d1 10<\/p>\n

                          \u24d0 [latex]{\\left(3a\\right)}^{0}[\/latex]
                          \n\u24d1 [latex]3{a}^{0}[\/latex]<\/p>\n

                          \u24d0 [latex]{\\left(-27{x}^{5}y\\right)}^{0}[\/latex]
                          \n\u24d1 [latex]-27{x}^{5}{y}^{0}[\/latex]<\/p>\n

                          \u24d0 1
                          \n\u24d1 \u221227x<\/em>5<\/sup><\/p>\n

                          \u24d0 [latex]{\\left(-92{y}^{8}z\\right)}^{0}[\/latex]
                          \n\u24d1 [latex]-92{y}^{8}{z}^{0}[\/latex]<\/p>\n

                          \u24d0 [latex]{15}^{0}[\/latex]
                          \n\u24d1 [latex]{15}^{1}[\/latex]<\/p>\n

                          \u24d0 1
                          \n\u24d1 15<\/p>\n

                          \u24d0 [latex]-{6}^{0}[\/latex]
                          \n\u24d1 [latex]-{6}^{1}[\/latex]<\/p>\n

                          [latex]2\\cdot {x}^{0}+5\\cdot {y}^{0}[\/latex]<\/p>\n

                          7<\/p>\n

                          [latex]8\\cdot {m}^{0}-4\\cdot {n}^{0}[\/latex]<\/p>\n

                          Simplify Expressions Using the Quotient to a Power Property<\/strong>
                          \nIn the following exercises, simplify.<\/p>\n

                          [latex]{\\left(\\Large\\frac{3}{2}\\normalsize\\right)}^{5}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{243}{32}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{4}{5}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{m}{6}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{m}^{3}}{216}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{p}{2}\\normalsize\\right)}^{5}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{x}{y}\\normalsize\\right)}^{10}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{x}^{10}}{{y}^{10}}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{a}{b}\\normalsize\\right)}^{8}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{a}{3b}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{a}^{2}}{9{b}^{2}}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{2x}{y}\\normalsize\\right)}^{4}[\/latex]<\/p>\n

                          Simplify Expressions by Applying Several Properties<\/strong>
                          \nIn the following exercises, simplify.<\/p>\n

                          [latex]\\Large\\frac{{\\left({x}^{2}\\right)}^{4}}{{x}^{5}}[\/latex]<\/p>\n

                          x<\/em>3<\/sup><\/p>\n

                          [latex]\\Large\\frac{{\\left({y}^{4}\\right)}^{3}}{{y}^{7}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left({u}^{3}\\right)}^{4}}{{u}^{10}}[\/latex]<\/p>\n

                          u<\/em>2<\/sup><\/p>\n

                          [latex]\\Large\\frac{{\\left({y}^{2}\\right)}^{5}}{{y}^{6}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{y}^{8}}{{\\left({y}^{5}\\right)}^{2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{y}^{2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{p}^{11}}{{\\left({p}^{5}\\right)}^{3}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{r}^{5}}{{r}^{4}\\cdot r}[\/latex]<\/p>\n

                          1<\/p>\n

                          [latex]\\Large\\frac{{a}^{3}\\cdot {a}^{4}}{{a}^{7}}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{{x}^{2}}{{x}^{8}}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{x}^{18}}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{u}{{u}^{10}}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{{a}^{4}\\cdot {a}^{6}}{{a}^{3}}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                          a<\/em>14<\/sup><\/p>\n

                          [latex]{\\left(\\Large\\frac{{x}^{3}\\cdot {x}^{8}}{{x}^{4}}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left({y}^{3}\\right)}^{5}}{{\\left({y}^{4}\\right)}^{3}}[\/latex]<\/p>\n

                          y<\/em>3<\/sup><\/p>\n

                          [latex]\\Large\\frac{{\\left({z}^{6}\\right)}^{2}}{{\\left({z}^{2}\\right)}^{4}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left({x}^{3}\\right)}^{6}}{{\\left({x}^{4}\\right)}^{7}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{x}^{10}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left({x}^{4}\\right)}^{8}}{{\\left({x}^{5}\\right)}^{7}}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{2{r}^{3}}{5s}\\normalsize\\right)}^{4}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{16{r}^{12}}{625{s}^{4}}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{3{m}^{2}}{4n}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                          [latex]{\\left(\\Large\\frac{3{y}^{2}\\cdot {y}^{5}}{{y}^{15}\\cdot {y}^{8}}\\normalsize\\right)}^{0}[\/latex]<\/p>\n

                          1<\/p>\n

                          [latex]{\\left(\\Large\\frac{15{z}^{4}\\cdot {z}^{9}}{0.3{z}^{2}}\\normalsize\\right)}^{0}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left({r}^{2}\\right)}^{5}{\\left({r}^{4}\\right)}^{2}}{{\\left({r}^{3}\\right)}^{7}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{r}^{3}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left({p}^{4}\\right)}^{2}{\\left({p}^{3}\\right)}^{5}}{{\\left({p}^{2}\\right)}^{9}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{\\left(3{x}^{4}\\right)}^{3}{\\left(2{x}^{3}\\right)}^{2}}{{\\left(6{x}^{5}\\right)}^{2}}[\/latex]<\/p>\n

                          3x<\/em>8<\/sup><\/p>\n

                          [latex]\\Large\\frac{{\\left(-2{y}^{3}\\right)}^{4}{\\left(3{y}^{4}\\right)}^{2}}{{\\left(-6{y}^{3}\\right)}^{2}}[\/latex]<\/p>\n

                          Divide Monomials<\/strong>
                          \nIn the following exercises, divide the monomials.<\/p>\n

                          [latex]48{b}^{8}\\div 6{b}^{2}[\/latex]<\/p>\n

                          8b<\/em>6<\/sup><\/p>\n

                          [latex]42{a}^{14}\\div 6{a}^{2}[\/latex]<\/p>\n

                          [latex]36{x}^{3}\\div \\left(-2{x}^{9}\\right)[\/latex]<\/p>\n

                          [latex]\\Large\\frac{-18}{{x}^{6}}[\/latex]<\/p>\n

                          [latex]20{u}^{8}\\div \\left(-4{u}^{6}\\right)[\/latex]<\/p>\n

                          [latex]\\Large\\frac{18{x}^{3}}{9{x}^{2}}[\/latex]<\/p>\n

                          2x<\/em><\/p>\n

                          [latex]\\Large\\frac{36{y}^{9}}{4{y}^{7}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{-35{x}^{7}}{-42{x}^{13}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{5}{6{x}^{6}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{18{x}^{5}}{-27{x}^{9}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{6{r}^{2}}{{s}^{8}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{24{p}^{7}q}{6{p}^{2}{q}^{5}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{8m{n}^{10}}{64m{n}^{4}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{n}^{6}}{8}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{10{a}^{4}b}{50{a}^{2}{b}^{6}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{-12{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}}[\/latex]<\/p>\n

                          [latex]-\\Large\\frac{4{y}^{6}}{5{x}^{2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{48{x}^{11}{y}^{9}{z}^{3}}{36{x}^{6}{y}^{8}{z}^{5}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{4z}{3{x}^{2}{y}^{3}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{\\left(10{u}^{2}v\\right)\\left(4{u}^{3}{v}^{6}\\right)}{5{u}^{9}{v}^{2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{\\left(6{m}^{2}n\\right)\\left(5{m}^{4}{n}^{3}\\right)}{3{m}^{10}{n}^{2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{10{n}^{2}}{{m}^{4}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{\\left(6{a}^{4}{b}^{3}\\right)\\left(4a{b}^{5}\\right)}{\\left(12{a}^{8}b\\right)\\left({a}^{3}b\\right)}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{\\left(4{u}^{5}{v}^{4}\\right)\\left(15{u}^{8}v\\right)}{\\left(12{u}^{3}v\\right)\\left({u}^{6}v\\right)}[\/latex]<\/p>\n

                          5u<\/em>4<\/sup>v<\/em>3<\/sup><\/p>\n

                          Mixed Practice<\/h2>\n

                          \u24d0 [latex]24{a}^{5}+2{a}^{5}[\/latex]
                          \n\u24d1 [latex]24{a}^{5}-2{a}^{5}[\/latex]
                          \n\u24d2 [latex]24{a}^{5}\\cdot 2{a}^{5}[\/latex]
                          \n\u24d3 [latex]24{a}^{5}\\div 2{a}^{5}[\/latex]<\/p>\n

                          \u24d0 [latex]15{n}^{10}+3{n}^{10}[\/latex]
                          \n\u24d1 [latex]15{n}^{10}-3{n}^{10}[\/latex]
                          \n\u24d2 [latex]15{n}^{10}\\cdot 3{n}^{10}[\/latex]
                          \n\u24d3 [latex]15{n}^{10}\\div 3{n}^{10}[\/latex]<\/p>\n

                          \u24d0 [latex]18{n}^{10}[\/latex]
                          \n\u24d1 [latex]12{n}^{10}[\/latex]
                          \n\u24d2 [latex]45{n}^{20}[\/latex]
                          \n\u24d3 [latex]5[\/latex]<\/p>\n

                          \u24d0 [latex]{p}^{4}\\cdot {p}^{6}[\/latex]
                          \n\u24d1 [latex]{\\left({p}^{4}\\right)}^{6}[\/latex]<\/p>\n

                          \u24d0 [latex]{q}^{5}\\cdot {q}^{3}[\/latex]
                          \n\u24d1 [latex]{\\left({q}^{5}\\right)}^{3}[\/latex]<\/p>\n

                          \u24d0 [latex]{q}^{8}[\/latex]
                          \n\u24d1 [latex]{q}^{15}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{{y}^{3}}{y}[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{y}{{y}^{3}}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{{z}^{6}}{{z}^{5}}[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{{z}^{5}}{{z}^{6}}[\/latex]<\/p>\n

                          \u24d0 [latex]z[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{1}{z}[\/latex]<\/p>\n

                          [latex]\\left(8{x}^{5}\\right)\\left(9x\\right)\\div 6{x}^{3}[\/latex]<\/p>\n

                          [latex]\\left(4{y}^{5}\\right)\\left(12{y}^{7}\\right)\\div 8{y}^{2}[\/latex]<\/p>\n

                          [latex]6{y}^{6}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{27{a}^{7}}{3{a}^{3}}\\normalsize +\\Large\\frac{54{a}^{9}}{9{a}^{5}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{32{c}^{11}}{4{c}^{5}}\\normalsize +\\Large\\frac{42{c}^{9}}{6{c}^{3}}[\/latex]<\/p>\n

                          [latex]15{c}^{6}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{32{y}^{5}}{8{y}^{2}}\\normalsize -\\Large\\frac{60{y}^{10}}{5{y}^{7}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{48{x}^{6}}{6{x}^{4}}\\normalsize -\\Large\\frac{35{x}^{9}}{7{x}^{7}}[\/latex]<\/p>\n

                          [latex]3{x}^{2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}\\normalsize -\\Large\\frac{72{r}^{2}{s}^{2}}{6s}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}\\normalsize -\\Large\\frac{45{y}^{2}{z}^{2}}{5y}[\/latex]<\/p>\n

                          [latex]y{z}^{2}[\/latex]<\/p>\n

                          Everyday Math<\/h2>\n

                          Memory<\/strong> One megabyte is approximately [latex]{10}^{6}[\/latex] bytes. One gigabyte is approximately [latex]{10}^{9}[\/latex] bytes. How many megabytes are in one gigabyte?<\/p>\n

                          Memory<\/strong> One megabyte is approximately [latex]{10}^{6}[\/latex] bytes. One terabyte is approximately [latex]{10}^{12}[\/latex] bytes. How many megabytes are in one terabyte?<\/p>\n

                          1,000,000<\/p>\n

                          Writing Exercises<\/h2>\n

                          Vic thinks the quotient [latex]\\Large\\frac{{x}^{20}}{{x}^{4}}[\/latex] simplifies to [latex]{x}^{5}[\/latex]. What is wrong with his reasoning?<\/p>\n

                          Mai simplifies the quotient [latex]\\Large\\frac{{y}^{3}}{y}[\/latex] by writing [latex]\\Large\\frac{{\\overline{)y}}^{3}}{\\overline{)y}}=3[\/latex]. What is wrong with her reasoning?<\/p>\n

                          Answers will vary.<\/p>\n

                          When Dimple simplified [latex]-{3}^{0}[\/latex] and [latex]{\\left(-3\\right)}^{0}[\/latex] she got the same answer. Explain how using the Order of Operations correctly gives different answers.<\/p>\n

                          Roxie thinks [latex]{n}^{0}[\/latex] simplifies to [latex]0[\/latex]. What would you say to convince Roxie she is wrong?<\/p>\n

                          Answers will vary.<\/p>\n

                          Self Check<\/h2>\n

                          \u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n

                          \".\"
                          \n\u24d1 On a scale of 1\u201310, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n

                          Practice Makes Perfect<\/h2>\n

                          Use the Definition of a Negative Exponent<\/strong>
                          \nIn the following exercises, simplify.<\/p>\n

                          [latex]{5}^{-3}[\/latex]<\/p>\n

                          [latex]{8}^{-2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{64}[\/latex]<\/p>\n

                          [latex]{3}^{-4}[\/latex]<\/p>\n

                          [latex]{2}^{-5}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{32}[\/latex]<\/p>\n

                          [latex]{7}^{-1}[\/latex]<\/p>\n

                          [latex]{10}^{-1}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{10}[\/latex]<\/p>\n

                          [latex]{2}^{-3}+{2}^{-2}[\/latex]<\/p>\n

                          [latex]{3}^{-2}+{3}^{-1}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{4}{9}[\/latex]<\/p>\n

                          [latex]{3}^{-1}+{4}^{-1}[\/latex]<\/p>\n

                          [latex]{10}^{-1}+{2}^{-1}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{3}{5}[\/latex]<\/p>\n

                          [latex]{10}^{0}-{10}^{-1}+{10}^{-2}[\/latex]<\/p>\n

                          [latex]{2}^{0}-{2}^{-1}+{2}^{-2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{3}{4}[\/latex]<\/p>\n

                          \u24d0 [latex]{\\left(-6\\right)}^{-2}[\/latex]
                          \n\u24d1 [latex]-{6}^{-2}[\/latex]<\/p>\n

                          \u24d0 [latex]{\\left(-8\\right)}^{-2}[\/latex]
                          \n\u24d1 [latex]-{8}^{-2}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{1}{64}[\/latex]
                          \n\u24d1 [latex]-\\Large\\frac{1}{64}[\/latex]<\/p>\n

                          \u24d0 [latex]{\\left(-10\\right)}^{-4}[\/latex]
                          \n\u24d1 [latex]-{10}^{-4}[\/latex]<\/p>\n

                          \u24d0 [latex]{\\left(-4\\right)}^{-6}[\/latex]
                          \n\u24d1 [latex]-{4}^{-6}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{1}{4096}[\/latex]
                          \n\u24d1 [latex]-\\Large\\frac{1}{4096}[\/latex]<\/p>\n

                          \u24d0 [latex]5\\cdot {2}^{-1}[\/latex]
                          \n\u24d1 [latex]{\\left(5\\cdot 2\\right)}^{-1}[\/latex]<\/p>\n

                          \u24d0 [latex]10\\cdot {3}^{-1}[\/latex]
                          \n\u24d1 [latex]{\\left(10\\cdot 3\\right)}^{-1}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{10}{3}[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{1}{30}[\/latex]<\/p>\n

                          \u24d0 [latex]4\\cdot {10}^{-3}[\/latex]
                          \n\u24d1 [latex]{\\left(4\\cdot 10\\right)}^{-3}[\/latex]<\/p>\n

                          \u24d0 [latex]3\\cdot {5}^{-2}[\/latex]
                          \n\u24d1 [latex]{\\left(3\\cdot 5\\right)}^{-2}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{3}{25}[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{1}{225}[\/latex]<\/p>\n

                          [latex]{n}^{-4}[\/latex]<\/p>\n

                          [latex]{p}^{-3}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{p}^{3}}[\/latex]<\/p>\n

                          [latex]{c}^{-10}[\/latex]<\/p>\n

                          [latex]{m}^{-5}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{m}^{5}}[\/latex]<\/p>\n

                          \u24d0 [latex]4{x}^{-1}[\/latex]
                          \n\u24d1 [latex]{\\left(4x\\right)}^{-1}[\/latex]
                          \n\u24d2 [latex]{\\left(-4x\\right)}^{-1}[\/latex]<\/p>\n

                          \u24d0 [latex]3{q}^{-1}[\/latex]
                          \n\u24d1 [latex]{\\left(3q\\right)}^{-1}[\/latex]
                          \n\u24d2 [latex]{\\left(-3q\\right)}^{-1}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{3}{q}[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{1}{3q}[\/latex]
                          \n\u24d2 [latex]-\\Large\\frac{1}{3q}[\/latex]<\/p>\n

                          \u24d0 [latex]6{m}^{-1}[\/latex]
                          \n\u24d1 [latex]{\\left(6m\\right)}^{-1}[\/latex]
                          \n\u24d2 [latex]{\\left(-6m\\right)}^{-1}[\/latex]<\/p>\n

                          \u24d0 [latex]10{k}^{-1}[\/latex]
                          \n\u24d1 [latex]{\\left(10k\\right)}^{-1}[\/latex]
                          \n\u24d2 [latex]{\\left(-10k\\right)}^{-1}[\/latex]<\/p>\n

                          \u24d0 [latex]\\Large\\frac{10}{k}[\/latex]
                          \n\u24d1 [latex]\\Large\\frac{1}{10k}[\/latex]
                          \n\u24d2 [latex]-\\Large\\frac{1}{10k}[\/latex]<\/p>\n

                          Simplify Expressions with Integer Exponents<\/strong>
                          \nIn the following exercises, simplify.<\/em><\/p>\n

                          [latex]{p}^{-4}\\cdot {p}^{8}[\/latex]<\/p>\n

                          [latex]{r}^{-2}\\cdot {r}^{5}[\/latex]<\/p>\n

                          r<\/em>3<\/sup><\/p>\n

                          [latex]{n}^{-10}\\cdot {n}^{2}[\/latex]<\/p>\n

                          [latex]{q}^{-8}\\cdot {q}^{3}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{q}^{5}}[\/latex]<\/p>\n

                          [latex]{k}^{-3}\\cdot {k}^{-2}[\/latex]<\/p>\n

                          [latex]{z}^{-6}\\cdot {z}^{-2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{z}^{8}}[\/latex]<\/p>\n

                          [latex]a\\cdot {a}^{-4}[\/latex]<\/p>\n

                          [latex]m\\cdot {m}^{-2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{m}[\/latex]<\/p>\n

                          [latex]{p}^{5}\\cdot {p}^{-2}\\cdot {p}^{-4}[\/latex]<\/p>\n

                          [latex]{x}^{4}\\cdot {x}^{-2}\\cdot {x}^{-3}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{x}[\/latex]<\/p>\n

                          [latex]{a}^{3}{b}^{-3}[\/latex]<\/p>\n

                          [latex]{u}^{2}{v}^{-2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{u}^{2}}{{v}^{2}}[\/latex]<\/p>\n

                          [latex]\\left({x}^{5}{y}^{-1}\\right)\\left({x}^{-10}{y}^{-3}\\right)[\/latex]<\/p>\n

                          [latex]\\left({a}^{3}{b}^{-3}\\right)\\left({a}^{-5}{b}^{-1}\\right)[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{a}^{2}{b}^{4}}[\/latex]<\/p>\n

                          [latex]\\left(u{v}^{-2}\\right)\\left({u}^{-5}{v}^{-4}\\right)[\/latex]<\/p>\n

                          [latex]\\left(p{q}^{-4}\\right)\\left({p}^{-6}{q}^{-3}\\right)[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{p}^{5}{q}^{7}}[\/latex]<\/p>\n

                          [latex]\\left(-2{r}^{-3}{s}^{9}\\right)\\left(6{r}^{4}{s}^{-5}\\right)[\/latex]<\/p>\n

                          [latex]\\left(-3{p}^{-5}{q}^{8}\\right)\\left(7{p}^{2}{q}^{-3}\\right)[\/latex]<\/p>\n

                          [latex]-\\Large\\frac{21{q}^{5}}{{p}^{3}}[\/latex]<\/p>\n

                          [latex]\\left(-6{m}^{-8}{n}^{-5}\\right)\\left(-9{m}^{4}{n}^{2}\\right)[\/latex]<\/p>\n

                          [latex]\\left(-8{a}^{-5}{b}^{-4}\\right)\\left(-4{a}^{2}{b}^{3}\\right)[\/latex]<\/p>\n

                          [latex]\\Large\\frac{32}{{a}^{3}b}[\/latex]<\/p>\n

                          [latex]{\\left({a}^{3}\\right)}^{-3}[\/latex]<\/p>\n

                          [latex]{\\left({q}^{10}\\right)}^{-10}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{q}^{100}}[\/latex]<\/p>\n

                          [latex]{\\left({n}^{2}\\right)}^{-1}[\/latex]<\/p>\n

                          [latex]{\\left({x}^{4}\\right)}^{-1}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{x}^{4}}[\/latex]<\/p>\n

                          [latex]{\\left({y}^{-5}\\right)}^{4}[\/latex]<\/p>\n

                          [latex]{\\left({p}^{-3}\\right)}^{2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{y}^{6}}[\/latex]<\/p>\n

                          [latex]{\\left({q}^{-5}\\right)}^{-2}[\/latex]<\/p>\n

                          [latex]{\\left({m}^{-2}\\right)}^{-3}[\/latex]<\/p>\n

                          m<\/em>6<\/sup><\/p>\n

                          [latex]{\\left(4{y}^{-3}\\right)}^{2}[\/latex]<\/p>\n

                          [latex]{\\left(3{q}^{-5}\\right)}^{2}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{9}{{q}^{10}}[\/latex]<\/p>\n

                          [latex]{\\left(10{p}^{-2}\\right)}^{-5}[\/latex]<\/p>\n

                          [latex]{\\left(2{n}^{-3}\\right)}^{-6}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{n}^{18}}{64}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{u}^{9}}{{u}^{-2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{b}^{5}}{{b}^{-3}}[\/latex]<\/p>\n

                          b<\/em>8<\/sup><\/p>\n

                          [latex]\\Large\\frac{{x}^{-6}}{{x}^{4}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{m}^{5}}{{m}^{-2}}[\/latex]<\/p>\n

                          m<\/em>7<\/sup><\/p>\n

                          [latex]\\Large\\frac{{q}^{3}}{{q}^{12}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{r}^{6}}{{r}^{9}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{1}{{r}^{3}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{n}^{-4}}{{n}^{-10}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{{p}^{-3}}{{p}^{-6}}[\/latex]<\/p>\n

                          p<\/em>3<\/sup><\/p>\n

                          Convert from Decimal Notation to Scientific Notation<\/strong>
                          \nIn the following exercises, write each number in scientific notation.<\/p>\n

                          45,000<\/p>\n

                          280,000<\/p>\n

                          2.8 \u00d7 105<\/sup><\/p>\n

                          8,750,000<\/p>\n

                          1,290,000<\/p>\n

                          1.29 \u00d7 106<\/sup><\/p>\n

                          0.036<\/p>\n

                          0.041<\/p>\n

                          4.1 \u00d7 10\u22122<\/sup><\/p>\n

                          0.00000924<\/p>\n

                          0.0000103<\/p>\n

                          1.03 \u00d7 10\u22125<\/sup><\/p>\n

                          The population of the United States on July 4, 2010 was almost [latex]310,000,000[\/latex].<\/p>\n

                          The population of the world on July 4, 2010 was more than [latex]6,850,000,000[\/latex].<\/p>\n

                          6.85 \u00d7 109<\/sup><\/p>\n

                          The average width of a human hair is [latex]0.0018[\/latex] centimeters.<\/p>\n

                          The probability of winning the [latex]2010[\/latex] Megamillions lottery is about [latex]0.0000000057[\/latex].<\/p>\n

                          5.7 \u00d7 10\u22129<\/sup><\/p>\n

                          Convert Scientific Notation to Decimal Form<\/strong>
                          \nIn the following exercises, convert each number to decimal form.<\/p>\n

                          [latex]4.1\\times {10}^{2}[\/latex]<\/p>\n

                          [latex]8.3\\times {10}^{2}[\/latex]<\/p>\n

                          830<\/p>\n

                          [latex]5.5\\times {10}^{8}[\/latex]<\/p>\n

                          [latex]1.6\\times {10}^{10}[\/latex]<\/p>\n

                          16,000,000,000<\/p>\n

                          [latex]3.5\\times {10}^{-2}[\/latex]<\/p>\n

                          [latex]2.8\\times {10}^{-2}[\/latex]<\/p>\n

                          0.028<\/p>\n

                          [latex]1.93\\times {10}^{-5}[\/latex]<\/p>\n

                          [latex]6.15\\times {10}^{-8}[\/latex]<\/p>\n

                          0.0000000615<\/p>\n

                          In 2010, the number of Facebook users each day who changed their status to \u2018engaged\u2019 was [latex]2\\times {10}^{4}[\/latex].<\/p>\n

                          At the start of 2012, the US federal budget had a deficit of more than [latex]{$1.5}\\times {10}^{13}[\/latex].<\/p>\n

                          $15,000,000,000,000<\/p>\n

                          The concentration of carbon dioxide in the atmosphere is [latex]3.9\\times {10}^{-4}[\/latex].<\/p>\n

                          The width of a proton is [latex]1\\times {10}^{-5}[\/latex] of the width of an atom.<\/p>\n

                          0.00001<\/p>\n

                          Multiply and Divide Using Scientific Notation<\/strong>
                          \nIn the following exercises, multiply or divide and write your answer in decimal form.<\/p>\n

                          [latex]\\left(2\\times {10}^{5}\\right)\\left(2\\times {10}^{-9}\\right)[\/latex]<\/p>\n

                          [latex]\\left(3\\times {10}^{2}\\right)\\left(1\\times {10}^{-5}\\right)[\/latex]<\/p>\n

                          0.003<\/p>\n

                          [latex]\\left(1.6\\times {10}^{-2}\\right)\\left(5.2\\times {10}^{-6}\\right)[\/latex]<\/p>\n

                          [latex]\\left(2.1\\times {10}^{-4}\\right)\\left(3.5\\times {10}^{-2}\\right)[\/latex]<\/p>\n

                          0.00000735<\/p>\n

                          [latex]\\Large\\frac{6\\times {10}^{4}}{3\\times {10}^{-2}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{8\\times {10}^{6}}{4\\times {10}^{-1}}[\/latex]<\/p>\n

                          200,000<\/p>\n

                          [latex]\\Large\\frac{7\\times {10}^{-2}}{1\\times {10}^{-8}}[\/latex]<\/p>\n

                          [latex]\\Large\\frac{5\\times {10}^{-3}}{1\\times {10}^{-10}}[\/latex]<\/p>\n

                          50,000,000<\/p>\n

                          Everyday Math<\/h2>\n

                          Calories<\/strong> In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by [latex]1.5[\/latex] trillion calories by the end of 2015.<\/p>\n

                            \n
                          1. \u24d0 Write [latex]1.5[\/latex] trillion in decimal notation.<\/li>\n
                          2. \u24d1 Write [latex]1.5[\/latex] trillion in scientific notation.<\/li>\n<\/ol>\n

                            Length of a year<\/strong> The difference between the calendar year and the astronomical year is [latex]0.000125[\/latex] day.<\/p>\n

                              \n
                            1. \u24d0 Write this number in scientific notation.<\/li>\n
                            2. \u24d1 How many years does it take for the difference to become 1 day?<\/li>\n<\/ol>\n
                                \n
                              1. \u24d0 1.25 \u00d7 10\u22124<\/sup><\/li>\n
                              2. \u24d0 8,000<\/li>\n<\/ol>\n

                                Calculator display<\/strong> Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator\u2019s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided [latex]1[\/latex] by [latex]2,598,960[\/latex] and saw the answer [latex]3.848\\times {10}^{-7}[\/latex]. Write the number in decimal notation.<\/p>\n

                                Calculator display<\/strong> Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator\u2019s display. To find the number of ways Barbara could make a collage with [latex]6[\/latex] of her [latex]50[\/latex] favorite photographs, she multiplied [latex]50\\cdot 49\\cdot 48\\cdot 47\\cdot 46\\cdot 45[\/latex]. Her calculator gave the answer [latex]1.1441304\\times {10}^{10}[\/latex]. Write the number in decimal notation.<\/p>\n

                                11,441,304,000<\/p>\n

                                Writing Exercises<\/h2>\n
                                  \n
                                1. \u24d0 Explain the meaning of the exponent in the expression [latex]{2}^{3}[\/latex].<\/li>\n
                                2. \u24d1 Explain the meaning of the exponent in the expression [latex]{2}^{-3}[\/latex]<\/li>\n<\/ol>\n

                                  When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?<\/p>\n

                                  Answers will vary.<\/p>\n

                                  Self Check<\/h2>\n

                                  \u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n

                                  \".\"
                                  \n\u24d1 After looking at the checklist, do you think you are well prepared for the next section? Why or why not?<\/p>\n

                                  <\/h2>\n

                                  Practice Makes Perfect<\/h2>\n

                                  Find the Greatest Common Factor of Two or More Expressions<\/strong>
                                  \nIn the following exercises, find the greatest common factor.<\/p>\n

                                  [latex]40,56[\/latex]<\/p>\n

                                  [latex]45,75[\/latex]<\/p>\n

                                  15<\/p>\n

                                  [latex]72,162[\/latex]<\/p>\n

                                  [latex]150,275[\/latex]<\/p>\n

                                  25<\/p>\n

                                  [latex]3x,12[\/latex]<\/p>\n

                                  [latex]4y,28[\/latex]<\/p>\n

                                  4<\/p>\n

                                  [latex]10a,50[\/latex]<\/p>\n

                                  [latex]5b,30[\/latex]<\/p>\n

                                  5<\/p>\n

                                  [latex]16y,24{y}^{2}[\/latex]<\/p>\n

                                  [latex]9x,15{x}^{2}[\/latex]<\/p>\n

                                  3x<\/em><\/p>\n

                                  [latex]18{m}^{3},36{m}^{2}[\/latex]<\/p>\n

                                  [latex]12{p}^{4},48{p}^{3}[\/latex]<\/p>\n

                                  12p<\/em>3<\/sup><\/p>\n

                                  [latex]10x,25{x}^{2},15{x}^{3}[\/latex]<\/p>\n

                                  [latex]18a,6{a}^{2},22{a}^{3}[\/latex]<\/p>\n

                                  2a<\/em><\/p>\n

                                  [latex]24u,6{u}^{2},30{u}^{3}[\/latex]<\/p>\n

                                  [latex]40y,10{y}^{2},90{y}^{3}[\/latex]<\/p>\n

                                  10y<\/em><\/p>\n

                                  [latex]15{a}^{4},9{a}^{5},21{a}^{6}[\/latex]<\/p>\n

                                  [latex]35{x}^{3},10{x}^{4},5{x}^{5}[\/latex]<\/p>\n

                                  5x<\/em>3<\/sup><\/p>\n

                                  [latex]27{y}^{2},45{y}^{3},9{y}^{4}[\/latex]<\/p>\n

                                  [latex]14{b}^{2},35{b}^{3},63{b}^{4}[\/latex]<\/p>\n

                                  7b<\/em>2<\/sup><\/p>\n

                                  Factor the Greatest Common Factor from a Polynomial<\/strong>
                                  \nIn the following exercises, factor the greatest common factor from each polynomial.<\/p>\n

                                  [latex]2x+8[\/latex]<\/p>\n

                                  [latex]5y+15[\/latex]<\/p>\n

                                  5(y<\/em> + 3)<\/p>\n

                                  [latex]3a - 24[\/latex]<\/p>\n

                                  [latex]4b - 20[\/latex]<\/p>\n

                                  4(b<\/em> \u2212 5)<\/p>\n

                                  [latex]9y - 9[\/latex]<\/p>\n

                                  [latex]7x - 7[\/latex]<\/p>\n

                                  7(x<\/em> \u2212 1)<\/p>\n

                                  [latex]5{m}^{2}+20m+35[\/latex]<\/p>\n

                                  [latex]3{n}^{2}+21n+12[\/latex]<\/p>\n

                                  3(n<\/em>2<\/sup> + 7n<\/em> + 4)<\/p>\n

                                  [latex]8{p}^{2}+32p+48[\/latex]<\/p>\n

                                  [latex]6{q}^{2}+30q+42[\/latex]<\/p>\n

                                  6(q<\/em>2<\/sup> + 5q<\/em> + 7)<\/p>\n

                                  [latex]8{q}^{2}+15q[\/latex]<\/p>\n

                                  [latex]9{c}^{2}+22c[\/latex]<\/p>\n

                                  c<\/em>(9c<\/em> + 22)<\/p>\n

                                  [latex]13{k}^{2}+5k[\/latex]<\/p>\n

                                  [latex]17{x}^{2}+7x[\/latex]<\/p>\n

                                  x<\/em>(17x<\/em> + 7)<\/p>\n

                                  [latex]5{c}^{2}+9c[\/latex]<\/p>\n

                                  [latex]4{q}^{2}+7q[\/latex]<\/p>\n

                                  q<\/em>(4q<\/em> + 7)<\/p>\n

                                  [latex]5{p}^{2}+25p[\/latex]<\/p>\n

                                  [latex]3{r}^{2}+27r[\/latex]<\/p>\n

                                  3r<\/em>(r<\/em> + 9)<\/p>\n

                                  [latex]24{q}^{2}-12q[\/latex]<\/p>\n

                                  [latex]30{u}^{2}-10u[\/latex]<\/p>\n

                                  10u<\/em>(3u<\/em> \u2212 1)<\/p>\n

                                  [latex]yz+4z[\/latex]<\/p>\n

                                  [latex]ab+8b[\/latex]<\/p>\n

                                  b<\/em>(a<\/em> + 8)<\/p>\n

                                  [latex]60x - 6{x}^{3}[\/latex]<\/p>\n

                                  [latex]55y - 11{y}^{4}[\/latex]<\/p>\n

                                  11y<\/em>(5 \u2212 y<\/em>3<\/sup>)<\/p>\n

                                  [latex]48{r}^{4}-12{r}^{3}[\/latex]<\/p>\n

                                  [latex]45{c}^{3}-15{c}^{2}[\/latex]<\/p>\n

                                  15c<\/em>2<\/sup>(3c<\/em> \u2212 1)<\/p>\n

                                  [latex]4{a}^{3}-4a{b}^{2}[\/latex]<\/p>\n

                                  [latex]6{c}^{3}-6c{d}^{2}[\/latex]<\/p>\n

                                  6c<\/em>(c<\/em>2<\/sup> \u2212 d<\/em>2<\/sup>)<\/p>\n

                                  [latex]30{u}^{3}+80{u}^{2}[\/latex]<\/p>\n

                                  [latex]48{x}^{3}+72{x}^{2}[\/latex]<\/p>\n

                                  24x<\/em>2<\/sup>(2x<\/em> + 3)<\/p>\n

                                  [latex]120{y}^{6}+48{y}^{4}[\/latex]<\/p>\n

                                  [latex]144{a}^{6}+90{a}^{3}[\/latex]<\/p>\n

                                  18a<\/em>3<\/sup>(8a<\/em>3<\/sup> + 5)<\/p>\n

                                  [latex]4{q}^{2}+24q+28[\/latex]<\/p>\n

                                  [latex]10{y}^{2}+50y+40[\/latex]<\/p>\n

                                  10(y<\/em>2<\/sup> + 5y<\/em> + 4)<\/p>\n

                                  [latex]15{z}^{2}-30z - 90[\/latex]<\/p>\n

                                  [latex]12{u}^{2}-36u - 108[\/latex]<\/p>\n

                                  12(u<\/em>2<\/sup> \u2212 3u<\/em> \u2212 9)<\/p>\n

                                  [latex]3{a}^{4}-24{a}^{3}+18{a}^{2}[\/latex]<\/p>\n

                                  [latex]5{p}^{4}-20{p}^{3}-15{p}^{2}[\/latex]<\/p>\n

                                  5p<\/em>2<\/sup>(p<\/em>2<\/sup> \u2212 4p<\/em> \u2212 3)<\/p>\n

                                  [latex]11{x}^{6}+44{x}^{5}-121{x}^{4}[\/latex]<\/p>\n

                                  [latex]8{c}^{5}+40{c}^{4}-56{c}^{3}[\/latex]<\/p>\n

                                  8c<\/em>3<\/sup>(c<\/em>2<\/sup> + 5c<\/em> \u2212 7)<\/p>\n

                                  [latex]-3n - 24[\/latex]<\/p>\n

                                  [latex]-7p - 84[\/latex]<\/p>\n

                                  \u22127(p<\/em> + 12)<\/p>\n

                                  [latex]-15{a}^{2}-40a[\/latex]<\/p>\n

                                  [latex]-18{b}^{2}-66b[\/latex]<\/p>\n

                                  \u22126b<\/em>(3b<\/em> + 11)<\/p>\n

                                  [latex]-10{y}^{3}+60{y}^{2}[\/latex]<\/p>\n

                                  [latex]-8{a}^{3}+32{a}^{2}[\/latex]<\/p>\n

                                  \u22128a<\/em>2<\/sup>(a<\/em> \u2212 4)<\/p>\n

                                  [latex]-4{u}^{5}+56{u}^{3}[\/latex]<\/p>\n

                                  [latex]-9{b}^{5}+63{b}^{3}[\/latex]<\/p>\n

                                  \u22129b<\/em>3<\/sup>(b<\/em>2<\/sup> \u2212 7)<\/p>\n

                                  Everyday Math<\/h2>\n

                                  Revenue<\/strong> A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of [latex]p[\/latex] dollars each is given by the polynomial [latex]-5{p}^{2}+150p[\/latex]. Factor the greatest common factor from this polynomial.<\/p>\n

                                  Height of a baseball<\/strong> The height of a baseball hit with velocity [latex]80[\/latex] feet\/second at [latex]4[\/latex] feet above ground level is [latex]-16{t}^{2}+80t+4[\/latex], with [latex]t=[\/latex] the number of seconds since it was hit. Factor the greatest common factor from this polynomial.<\/p>\n

                                  \u22124(4t<\/em>2<\/sup> \u2212 20t<\/em> \u2212 1)<\/p>\n

                                  Writing Exercises<\/h2>\n

                                  The greatest common factor of [latex]36[\/latex] and [latex]60[\/latex] is [latex]12[\/latex]. Explain what this means.<\/p>\n

                                  What is the GCF of [latex]{y}^{4}[\/latex] , [latex]{y}^{5}[\/latex] , and [latex]{y}^{10}[\/latex] ? Write a general rule that tells how to find the GCF of [latex]{y}^{\\text{a}}[\/latex] , [latex]{y}^{\\text{b}}[\/latex] , and [latex]{y}^{\\text{c}}[\/latex] .<\/p>\n

                                  Self Check<\/h2>\n

                                  \u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n

                                  \".\"
                                  \n\u24d1 Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?<\/p>\n

                                  Chapter Review Exercises<\/h1>\n

                                  Add and Subtract Polynomials<\/h2>\n

                                  Identify Polynomials, Monomials, Binomials and Trinomials<\/strong>
                                  \nIn the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.<\/p>\n

                                  [latex]{y}^{2}+8y - 20[\/latex]<\/p>\n

                                  trinomial<\/p>\n

                                  [latex]-6{a}^{4}[\/latex]<\/p>\n

                                  [latex]9{x}^{3}-1[\/latex]<\/p>\n

                                  binomial<\/p>\n

                                  [latex]{n}^{3}-3{n}^{2}+3n - 1[\/latex]<\/p>\n

                                  Determine the Degree of Polynomials<\/strong>
                                  \nIn the following exercises, determine the degree of each polynomial.<\/p>\n

                                  [latex]16{x}^{2}-40x - 25[\/latex]<\/p>\n

                                  2<\/p>\n

                                  [latex]5m+9[\/latex]<\/p>\n

                                  [latex]-15[\/latex]<\/p>\n

                                  0<\/p>\n

                                  [latex]{y}^{2}+6{y}^{3}+9{y}^{4}[\/latex]<\/p>\n

                                  Add and Subtract Monomials<\/strong>
                                  \nIn the following exercises, add or subtract the monomials.<\/p>\n

                                  [latex]4p+11p[\/latex]<\/p>\n

                                  15p<\/em><\/p>\n

                                  [latex]-8{y}^{3}-5{y}^{3}[\/latex]<\/p>\n

                                  Add [latex]4{n}^{5},\\text{-}{n}^{5},-6{n}^{5}[\/latex]<\/p>\n

                                  \u22123n<\/em>5<\/sup><\/p>\n

                                  Subtract [latex]10{x}^{2}[\/latex] from [latex]3{x}^{2}[\/latex]<\/p>\n

                                  Add and Subtract Polynomials<\/strong>
                                  \nIn the following exercises, add or subtract the polynomials.<\/p>\n

                                  [latex]\\left(4{a}^{2}+9a - 11\\right)+\\left(6{a}^{2}-5a+10\\right)[\/latex]<\/p>\n

                                  10a<\/em>2<\/sup> + 4a<\/em> \u2212 1<\/p>\n

                                  [latex]\\left(8{m}^{2}+12m - 5\\right)-\\left(2{m}^{2}-7m - 1\\right)[\/latex]<\/p>\n

                                  [latex]\\left({y}^{2}-3y+12\\right)+\\left(5{y}^{2}-9\\right)[\/latex]<\/p>\n

                                  6y<\/em>2<\/sup> \u2212 3y<\/em> + 3<\/p>\n

                                  [latex]\\left(5{u}^{2}+8u\\right)-\\left(4u - 7\\right)[\/latex]<\/p>\n

                                  Find the sum of [latex]8{q}^{3}-27[\/latex] and [latex]{q}^{2}+6q - 2[\/latex]<\/p>\n

                                  8q<\/em>3<\/sup> + q<\/em>2<\/sup> + 6q<\/em> \u2212 29<\/p>\n

                                  Find the difference of [latex]{x}^{2}+6x+8[\/latex] and [latex]{x}^{2}-8x+15[\/latex]<\/p>\n

                                  Evaluate a Polynomial for a Given Value of the Variable<\/strong>
                                  \nIn the following exercises, evaluate each polynomial for the given value.<\/p>\n

                                  [latex]200x-\\Large\\frac{1}{5}\\normalsize{x}^{2}[\/latex] when [latex]x=5[\/latex]<\/p>\n

                                  995<\/p>\n

                                  [latex]200x-\\Large\\frac{1}{5}\\normalsize{x}^{2}[\/latex] when [latex]x=0[\/latex]<\/p>\n

                                  [latex]200x-\\Large\\frac{1}{5}\\normalsize{x}^{2}[\/latex] when [latex]x=15[\/latex]<\/p>\n

                                  2,955<\/p>\n

                                  [latex]5+40x-\\Large\\frac{1}{2}\\normalsize{x}^{2}[\/latex] when [latex]x=10[\/latex]<\/p>\n

                                  [latex]5+40x-\\Large\\frac{1}{2}\\normalsize{x}^{2}[\/latex] when [latex]x=-4[\/latex]<\/p>\n

                                  \u2212163<\/p>\n

                                  [latex]5+40x-\\Large\\frac{1}{2}\\normalsize{x}^{2}[\/latex] when [latex]x=0[\/latex]<\/p>\n

                                  A pair of glasses is dropped off a bridge [latex]640[\/latex] feet above a river. The polynomial [latex]-16{t}^{2}+640[\/latex] gives the height of the glasses [latex]t[\/latex] seconds after they were dropped. Find the height of the glasses when [latex]t=6[\/latex].<\/p>\n

                                  64 feet<\/p>\n

                                  The fuel efficiency (in miles per gallon) of a bus going at a speed of [latex]x[\/latex] miles per hour is given by the polynomial [latex]-\\Large\\frac{1}{160}\\normalsize{x}^{2}+\\Large\\frac{1}{2}\\normalsize x[\/latex]. Find the fuel efficiency when [latex]x=20[\/latex] mph.<\/p>\n

                                  Use Multiplication Properties of Exponents<\/h2>\n

                                  Simplify Expressions with Exponents<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]{6}^{3}[\/latex]<\/p>\n

                                  216<\/p>\n

                                  [latex]{\\left(\\Large\\frac{1}{2}\\normalsize\\right)}^{4}[\/latex]<\/p>\n

                                  [latex]{\\left(-0.5\\right)}^{2}[\/latex]<\/p>\n

                                  0.25<\/p>\n

                                  [latex]-{3}^{2}[\/latex]<\/p>\n

                                  Simplify Expressions Using the Product Property of Exponents<\/strong>
                                  \nIn the following exercises, simplify each expression.<\/p>\n

                                  [latex]{p}^{3}\\cdot {p}^{10}[\/latex]<\/p>\n

                                  p<\/em>13<\/sup><\/p>\n

                                  [latex]2\\cdot {2}^{6}[\/latex]<\/p>\n

                                  [latex]a\\cdot {a}^{2}\\cdot {a}^{3}[\/latex]<\/p>\n

                                  a<\/em>6<\/sup><\/p>\n

                                  [latex]x\\cdot {x}^{8}[\/latex]<\/p>\n

                                  Simplify Expressions Using the Power Property of Exponents<\/strong>
                                  \nIn the following exercises, simplify each expression.<\/p>\n

                                  [latex]{\\left({y}^{4}\\right)}^{3}[\/latex]<\/p>\n

                                  y<\/em>12<\/sup><\/p>\n

                                  [latex]{\\left({r}^{3}\\right)}^{2}[\/latex]<\/p>\n

                                  [latex]{\\left({3}^{2}\\right)}^{5}[\/latex]<\/p>\n

                                  310<\/sup><\/p>\n

                                  [latex]{\\left({a}^{10}\\right)}^{y}[\/latex]<\/p>\n

                                  Simplify Expressions Using the Product to a Power Property<\/strong>
                                  \nIn the following exercises, simplify each expression.<\/p>\n

                                  [latex]{\\left(8n\\right)}^{2}[\/latex]<\/p>\n

                                  64n<\/em>2<\/sup><\/p>\n

                                  [latex]{\\left(-5x\\right)}^{3}[\/latex]<\/p>\n

                                  [latex]{\\left(2ab\\right)}^{8}[\/latex]<\/p>\n

                                  256a<\/em>8<\/sup>b<\/em>8<\/sup><\/p>\n

                                  [latex]{\\left(-10mnp\\right)}^{4}[\/latex]<\/p>\n

                                  Simplify Expressions by Applying Several Properties<\/strong>
                                  \nIn the following exercises, simplify each expression.<\/p>\n

                                  [latex]{\\left(3{a}^{5}\\right)}^{3}[\/latex]<\/p>\n

                                  27a<\/em>15<\/sup><\/p>\n

                                  [latex]{\\left(4y\\right)}^{2}\\left(8y\\right)[\/latex]<\/p>\n

                                  [latex]{\\left({x}^{3}\\right)}^{5}{\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n

                                  x<\/em>21<\/sup><\/p>\n

                                  [latex]{\\left(5s{t}^{2}\\right)}^{3}{\\left(2{s}^{3}{t}^{4}\\right)}^{2}[\/latex]<\/p>\n

                                  Multiply Monomials<\/strong>
                                  \nIn the following exercises, multiply the monomials.<\/p>\n

                                  [latex]\\left(-6{p}^{4}\\right)\\left(9p\\right)[\/latex]<\/p>\n

                                  \u221254p<\/em>5<\/sup><\/p>\n

                                  [latex]\\left(\\Large\\frac{1}{3}\\normalsize{c}^{2}\\right)\\left(30{c}^{8}\\right)[\/latex]<\/p>\n

                                  [latex]\\left(8{x}^{2}{y}^{5}\\right)\\left(7x{y}^{6}\\right)[\/latex]<\/p>\n

                                  56x<\/em>3<\/sup>y<\/em>11<\/sup><\/p>\n

                                  [latex]\\left(\\Large\\frac{2}{3}\\normalsize{m}^{3}{n}^{6}\\right)\\left(\\Large\\frac{1}{6}\\normalsize{m}^{4}{n}^{4}\\right)[\/latex]<\/p>\n

                                  Multiply Polynomials<\/h2>\n

                                  Multiply a Polynomial by a Monomial<\/strong>
                                  \nIn the following exercises, multiply.<\/p>\n

                                  [latex]7\\left(10-x\\right)[\/latex]<\/p>\n

                                  70 \u2212 7x<\/em><\/p>\n

                                  [latex]{a}^{2}\\left({a}^{2}-9a - 36\\right)[\/latex]<\/p>\n

                                  [latex]-5y\\left(125{y}^{3}-1\\right)[\/latex]<\/p>\n

                                  \u2212625y<\/em>4<\/sup> + 5y<\/em><\/p>\n

                                  [latex]\\left(4n - 5\\right)\\left(2{n}^{3}\\right)[\/latex]<\/p>\n

                                  Multiply a Binomial by a Binomial<\/strong>
                                  \nIn the following exercises, multiply the binomials using various methods.<\/p>\n

                                  [latex]\\left(a+5\\right)\\left(a+2\\right)[\/latex]<\/p>\n

                                  a<\/em>2<\/sup> + 7a<\/em> + 10<\/p>\n

                                  [latex]\\left(y - 4\\right)\\left(y+12\\right)[\/latex]<\/p>\n

                                  [latex]\\left(3x+1\\right)\\left(2x - 7\\right)[\/latex]<\/p>\n

                                  6x<\/em>2<\/sup> \u2212 19x<\/em> \u2212 7<\/p>\n

                                  [latex]\\left(6p - 11\\right)\\left(3p - 10\\right)[\/latex]<\/p>\n

                                  [latex]\\left(n+8\\right)\\left(n+1\\right)[\/latex]<\/p>\n

                                  n<\/em>2<\/sup> + 9n<\/em> + 8<\/p>\n

                                  [latex]\\left(k+6\\right)\\left(k - 9\\right)[\/latex]<\/p>\n

                                  [latex]\\left(5u - 3\\right)\\left(u+8\\right)[\/latex]<\/p>\n

                                  5u<\/em>2<\/sup> + 37u<\/em> \u2212 24<\/p>\n

                                  [latex]\\left(2y - 9\\right)\\left(5y - 7\\right)[\/latex]<\/p>\n

                                  [latex]\\left(p+4\\right)\\left(p+7\\right)[\/latex]<\/p>\n

                                  p<\/em>2<\/sup> + 11p<\/em> + 28<\/p>\n

                                  [latex]\\left(x - 8\\right)\\left(x+9\\right)[\/latex]<\/p>\n

                                  [latex]\\left(3c+1\\right)\\left(9c - 4\\right)[\/latex]<\/p>\n

                                  27c<\/em>2<\/sup> \u2212 3c<\/em> \u2212 4<\/p>\n

                                  [latex]\\left(10a - 1\\right)\\left(3a - 3\\right)[\/latex]<\/p>\n

                                  Multiply a Trinomial by a Binomial<\/strong>
                                  \nIn the following exercises, multiply using any method.<\/p>\n

                                  [latex]\\left(x+1\\right)\\left({x}^{2}-3x - 21\\right)[\/latex]<\/p>\n

                                  x<\/em>3<\/sup> \u2212 2x<\/em>2<\/sup> \u2212 24x<\/em> \u2212 21<\/p>\n

                                  [latex]\\left(5b - 2\\right)\\left(3{b}^{2}+b - 9\\right)[\/latex]<\/p>\n

                                  [latex]\\left(m+6\\right)\\left({m}^{2}-7m - 30\\right)[\/latex]<\/p>\n

                                  m<\/em>3<\/sup> \u2212 m<\/em>2<\/sup> \u2212 72m<\/em> \u2212 180<\/p>\n

                                  [latex]\\left(4y - 1\\right)\\left(6{y}^{2}-12y+5\\right)[\/latex]<\/p>\n

                                  Divide Monomials<\/h2>\n

                                  Simplify Expressions Using the Quotient Property of Exponents<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]\\Large\\frac{{2}^{8}}{{2}^{2}}[\/latex]<\/p>\n

                                  26<\/sup>or<\/em> 64<\/p>\n

                                  [latex]\\Large\\frac{{a}^{6}}{a}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{{n}^{3}}{{n}^{12}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{1}{{n}^{9}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{x}{{x}^{5}}[\/latex]<\/p>\n

                                  Simplify Expressions with Zero Exponents<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]{3}^{0}[\/latex]<\/p>\n

                                  1<\/p>\n

                                  [latex]{y}^{0}[\/latex]<\/p>\n

                                  [latex]{\\left(14t\\right)}^{0}[\/latex]<\/p>\n

                                  1<\/p>\n

                                  [latex]12{a}^{0}-15{b}^{0}[\/latex]<\/p>\n

                                  Simplify Expressions Using the Quotient to a Power Property<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]{\\left(\\Large\\frac{3}{5}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{9}{25}[\/latex]<\/p>\n

                                  [latex]{\\left(\\Large\\frac{x}{2}\\normalsize\\right)}^{5}[\/latex]<\/p>\n

                                  [latex]{\\left(\\Large\\frac{5m}{n}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{125{m}^{3}}{{n}^{3}}[\/latex]<\/p>\n

                                  [latex]{\\left(\\Large\\frac{s}{10t}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                                  Simplify Expressions by Applying Several Properties<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]\\Large\\frac{{\\left({a}^{3}\\right)}^{2}}{{a}^{4}}[\/latex]<\/p>\n

                                  a<\/em>2<\/sup><\/p>\n

                                  [latex]\\Large\\frac{{u}^{3}}{{u}^{2}\\normalsize\\cdot {u}^{4}}[\/latex]<\/p>\n

                                  [latex]{\\left(\\Large\\frac{x}{{x}^{9}}\\normalsize\\right)}^{5}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{1}{{x}^{40}}[\/latex]<\/p>\n

                                  [latex]{\\left(\\Large\\frac{{p}^{4}\\cdot {p}^{5}}{{p}^{3}}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{{\\left({n}^{5}\\right)}^{3}}{{\\left({n}^{2}\\right)}^{8}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{1}{n}[\/latex]<\/p>\n

                                  [latex]{\\left(\\Large\\frac{5{s}^{2}}{4t}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                                  Divide Monomials<\/strong>
                                  \nIn the following exercises, divide the monomials.<\/p>\n

                                  [latex]72{p}^{12}\\div 8{p}^{3}[\/latex]<\/p>\n

                                  9p<\/em>9<\/sup><\/p>\n

                                  [latex]-26{a}^{8}\\div \\left(2{a}^{2}\\right)[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{45{y}^{6}}{-15{y}^{10}}[\/latex]<\/p>\n

                                  [latex]-\\Large\\frac{3}{{y}^{4}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{-30{x}^{8}}{-36{x}^{9}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{28{a}^{9}b}{7{a}^{4}{b}^{3}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{4{a}^{5}}{{b}^{2}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{11{u}^{6}{v}^{3}}{55{u}^{2}{v}^{8}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{\\left(5{m}^{9}{n}^{3}\\right)\\left(8{m}^{3}{n}^{2}\\right)}{\\left(10m{n}^{4}\\right)\\left({m}^{2}{n}^{5}\\right)}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{4{m}^{9}}{{n}^{4}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{42{r}^{2}{s}^{4}}{6r{s}^{3}}\\normalsize -\\Large\\frac{54r{s}^{2}}{9s}[\/latex]<\/p>\n

                                  Integer Exponents and Scientific Notation<\/h2>\n

                                  Use the Definition of a Negative Exponent<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]{6}^{-2}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{1}{36}[\/latex]<\/p>\n

                                  [latex]{\\left(-10\\right)}^{-3}[\/latex]<\/p>\n

                                  [latex]5\\cdot {2}^{-4}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{5}{16}[\/latex]<\/p>\n

                                  [latex]{\\left(8n\\right)}^{-1}[\/latex]<\/p>\n

                                  Simplify Expressions with Integer Exponents<\/strong>
                                  \nIn the following exercises, simplify.<\/p>\n

                                  [latex]{x}^{-3}\\cdot {x}^{9}[\/latex]<\/p>\n

                                  x<\/em>6<\/sup><\/p>\n

                                  [latex]{r}^{-5}\\cdot {r}^{-4}[\/latex]<\/p>\n

                                  [latex]\\left(u{v}^{-3}\\right)\\left({u}^{-4}{v}^{-2}\\right)[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{1}{{u}^{3}{v}^{5}}[\/latex]<\/p>\n

                                  [latex]{\\left({m}^{5}\\right)}^{-1}[\/latex]<\/p>\n

                                  [latex]{\\left({k}^{-2}\\right)}^{-3}[\/latex]<\/p>\n

                                  k<\/em>6<\/sup><\/p>\n

                                  [latex]\\Large\\frac{{q}^{4}}{{q}^{20}}[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{{b}^{8}}{{b}^{-2}}[\/latex]<\/p>\n

                                  b<\/em>10<\/sup><\/p>\n

                                  [latex]\\Large\\frac{{n}^{-3}}{{n}^{-5}}[\/latex]<\/p>\n

                                  Convert from Decimal Notation to Scientific Notation<\/strong>
                                  \nIn the following exercises, write each number in scientific notation.<\/p>\n

                                  [latex]5,300,000[\/latex]<\/p>\n

                                  5.3 \u00d7 106<\/sup><\/p>\n

                                  [latex]0.00814[\/latex]<\/p>\n

                                  The thickness of a piece of paper is about [latex]0.097[\/latex] millimeter.<\/p>\n

                                  9.7 \u00d7 10\u22122<\/sup><\/p>\n

                                  According to www.cleanair.com, U.S. businesses use about [latex]21,000,000[\/latex] tons of paper per year.<\/p>\n

                                  Convert Scientific Notation to Decimal Form<\/strong>
                                  \nIn the following exercises, convert each number to decimal form.<\/p>\n

                                  [latex]2.9\\times {10}^{4}[\/latex]<\/p>\n

                                  29,000<\/p>\n

                                  [latex]1.5\\times {10}^{8}[\/latex]<\/p>\n

                                  [latex]3.75\\times {10}^{-1}[\/latex]<\/p>\n

                                  375<\/p>\n

                                  [latex]9.413\\times {10}^{-5}[\/latex]<\/p>\n

                                  Multiply and Divide Using Scientific Notation<\/strong>
                                  \nIn the following exercises, multiply and write your answer in decimal form.<\/p>\n

                                  [latex]\\left(3\\times {10}^{7}\\right)\\left(2\\times {10}^{-4}\\right)[\/latex]<\/p>\n

                                  6,000<\/p>\n

                                  [latex]\\left(1.5\\times {10}^{-3}\\right)\\left(4.8\\times {10}^{-1}\\right)[\/latex]<\/p>\n

                                  [latex]\\Large\\frac{6\\times {10}^{9}}{2\\times {10}^{-1}}[\/latex]<\/p>\n

                                  30,000,000,000<\/p>\n

                                  [latex]\\Large\\frac{9\\times {10}^{-3}}{1\\times {10}^{-6}}[\/latex]<\/p>\n

                                  Introduction to Factoring Polynomials<\/h2>\n

                                  Find the Greatest Common Factor of Two or More Expressions<\/strong>
                                  \nIn the following exercises, find the greatest common factor.<\/p>\n

                                  [latex]5n,45[\/latex]<\/p>\n

                                  5<\/p>\n

                                  [latex]8a,72[\/latex]<\/p>\n

                                  [latex]12{x}^{2},20{x}^{3},36{x}^{4}[\/latex]<\/p>\n

                                  4x<\/em>2<\/sup><\/p>\n

                                  [latex]9{y}^{4},21{y}^{5},15{y}^{6}[\/latex]<\/p>\n

                                  Factor the Greatest Common Factor from a Polynomial<\/strong>
                                  \nIn the following exercises, factor the greatest common factor from each polynomial.<\/p>\n

                                  [latex]16u - 24[\/latex]<\/p>\n

                                  8(2u<\/em> \u2212 3)<\/p>\n

                                  [latex]15r+35[\/latex]<\/p>\n

                                  [latex]6{p}^{2}+6p[\/latex]<\/p>\n

                                  6p<\/em>(p<\/em> + 1)<\/p>\n

                                  [latex]10{c}^{2}-10c[\/latex]<\/p>\n

                                  [latex]-9{a}^{5}-9{a}^{3}[\/latex]<\/p>\n

                                  \u22129a<\/em>3<\/sup>(a<\/em>2<\/sup> + 1)<\/p>\n

                                  [latex]-7{x}^{8}-28{x}^{3}[\/latex]<\/p>\n

                                  [latex]5{y}^{2}-55y+45[\/latex]<\/p>\n

                                  5(y<\/em>2<\/sup> \u2212 11y<\/em> + 9)<\/p>\n

                                  [latex]2{q}^{5}-16{q}^{3}+30{q}^{2}[\/latex]<\/p>\n

                                  Chapter Practice Test<\/h1>\n

                                  For the polynomial [latex]8{y}^{4}-3{y}^{2}+1[\/latex]<\/p>\n

                                    \n
                                  1. \u24d0 Is it a monomial, binomial, or trinomial?<\/li>\n
                                  2. \u24d1 What is its degree?<\/li>\n<\/ol>\n
                                      \n
                                    1. \u24d0 trinomial<\/li>\n
                                    2. \u24d1 4<\/li>\n<\/ol>\n

                                      In the following exercises, simplify each expression.<\/p>\n

                                      [latex]\\left(5{a}^{2}+2a - 12\\right)+\\left(9{a}^{2}+8a - 4\\right)[\/latex]<\/p>\n

                                      [latex]\\left(10{x}^{2}-3x+5\\right)-\\left(4{x}^{2}-6\\right)[\/latex]<\/p>\n

                                      6x<\/em>2<\/sup> \u2212 3x<\/em> + 11<\/p>\n

                                      [latex]{\\left(-\\Large\\frac{3}{4}\\normalsize\\right)}^{3}[\/latex]<\/p>\n

                                      [latex]n\\cdot {n}^{4}[\/latex]<\/p>\n

                                      n<\/em>5<\/sup><\/p>\n

                                      [latex]{\\left(10{p}^{3}{q}^{5}\\right)}^{2}[\/latex]<\/p>\n

                                      [latex]\\left(8x{y}^{3}\\right)\\left(-6{x}^{4}{y}^{6}\\right)[\/latex]<\/p>\n

                                      \u221248x<\/em>5<\/sup>y<\/em>9<\/sup><\/p>\n

                                      [latex]4u\\left({u}^{2}-9u+1\\right)[\/latex]<\/p>\n

                                      [latex]\\left(s+8\\right)\\left(s+9\\right)[\/latex]<\/p>\n

                                      s<\/em>2<\/sup> + 17s<\/em> + 72<\/p>\n

                                      [latex]\\left(m+3\\right)\\left(7m - 2\\right)[\/latex]<\/p>\n

                                      [latex]\\left(11a - 6\\right)\\left(5a - 1\\right)[\/latex]<\/p>\n

                                      55a<\/em>2<\/sup> \u2212 41a<\/em> + 6<\/p>\n

                                      [latex]\\left(n - 8\\right)\\left({n}^{2}-4n+11\\right)[\/latex]<\/p>\n

                                      [latex]\\left(4a+9b\\right)\\left(6a - 5b\\right)[\/latex]<\/p>\n

                                      24a<\/em>2<\/sup> + 34ab<\/em> \u2212 45b<\/em>2<\/sup><\/p>\n

                                      [latex]\\Large\\frac{{5}^{6}}{{5}^{8}}[\/latex]<\/p>\n

                                      [latex]{\\left(\\Large\\frac{{x}^{3}\\cdot {x}^{9}}{{x}^{5}}\\normalsize\\right)}^{2}[\/latex]<\/p>\n

                                      x<\/em>14<\/sup><\/p>\n

                                      [latex]{\\left(47{a}^{18}{b}^{23}{c}^{5}\\right)}^{0}[\/latex]<\/p>\n

                                      [latex]\\Large\\frac{24{r}^{3}s}{6{r}^{2}{s}^{7}}[\/latex]<\/p>\n

                                      [latex]\\Large\\frac{4r}{{s}^{6}}[\/latex]<\/p>\n

                                      [latex]\\Large\\frac{8{y}^{2}-16y+20}{4y}[\/latex]<\/p>\n

                                      [latex]\\left(15x{y}^{3}-35{x}^{2}y\\right)\\div 5xy[\/latex]<\/p>\n

                                      3y<\/em>2<\/sup> \u2212 7x<\/em><\/p>\n

                                      [latex]{4}^{-1}[\/latex]<\/p>\n

                                      [latex]{\\left(2y\\right)}^{-3}[\/latex]<\/p>\n

                                      [latex]\\Large\\frac{1}{2y}[\/latex]<\/p>\n

                                      [latex]{p}^{-3}\\cdot {p}^{-8}[\/latex]<\/p>\n

                                      [latex]\\Large\\frac{{x}^{4}}{{x}^{-5}}[\/latex]<\/p>\n

                                      x<\/em>9<\/sup><\/p>\n

                                      [latex]\\left(2.4\\times {10}^{8}\\right)\\left(2\\times {10}^{-5}\\right)[\/latex]<\/p>\n

                                      In the following exercises, factor the greatest common factor from each polynomial.<\/p>\n

                                      [latex]80{a}^{3}+120{a}^{2}+40a[\/latex]<\/p>\n

                                      [latex]-6{x}^{2}-30x[\/latex]<\/p>\n

                                      \u22126x<\/em>(x<\/em> + 5)<\/p>\n

                                      Convert [latex]5.25\\times {10}^{-4}[\/latex] to decimal form.<\/p>\n

                                      0.000525<\/p>\n

                                      In the following exercises, simplify, and write your answer in decimal form.<\/p>\n

                                      [latex]\\Large\\frac{9\\times {10}^{4}}{3\\times {10}^{-1}}[\/latex]<\/p>\n

                                      3 \u00d7 105<\/sup><\/p>\n

                                      A hiker drops a pebble from a bridge [latex]240[\/latex] feet above a canyon. The polynomial [latex]-16{t}^{2}+240[\/latex] gives the height of the pebble [latex]t[\/latex] seconds a after it was dropped. Find the height when [latex]t=3[\/latex].<\/p>\n

                                      According to www.cleanair.org, the amount of trash generated in the US in one year averages out to [latex]112,000[\/latex] pounds of trash per person. Write this number in scientific notation.<\/p>\n

                                      <\/h2>\n

                                      Contribute!<\/h2>\n
                                      Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n

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