{"id":41,"date":"2023-06-21T13:22:27","date_gmt":"2023-06-21T13:22:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-and-simplify-algebraic-expressions\/"},"modified":"2023-07-04T04:53:27","modified_gmt":"2023-07-04T04:53:27","slug":"evaluate-and-simplify-algebraic-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-and-simplify-algebraic-expressions\/","title":{"raw":"\u25aa   Evaluate and Simplify Algebraic Expressions","rendered":"\u25aa   Evaluate and Simplify Algebraic Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>List the constants and variables in an algebraic expression.<\/li>\r\n \t<li>Evaluate an algebraic expression.<\/li>\r\n \t<li>Use an algebraic formula.<\/li>\r\n<\/ul>\r\n<\/div>\r\nSo far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as [latex]x+5,\\frac{4}{3}\\pi {r}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5, 5[\/latex] is called a <strong>constant<\/strong> because it does not vary and <em>x<\/em> is called a <strong>variable<\/strong> because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.\r\n\r\nWe have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right) &amp;&amp; x^{5}=x\\cdot x\\cdot x\\cdot x\\cdot x \\\\ &amp;\\left(2\\cdot7\\right)^{3}=\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right) &amp;&amp; \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\nIn each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.\r\n\r\nAny variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Describing Algebraic Expressions<\/h3>\r\nList the constants and variables for each algebraic expression.\r\n<ol>\r\n \t<li><em>x<\/em> + 5<\/li>\r\n \t<li>[latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"790423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790423\"]\r\n<table summary=\"A table with four rows and three columns. The first entry of the first row is empty, but the second entry reads: Constants, and the third reads: Variables. The first entry of the second row reads: x plus five. The second column entry reads: five. The third column entry reads: x. The first entry of the third row reads: four-thirds pi times r cubed. The second column entry reads: four-thirds, pi. The third column entry reads: r. The first entry of the fourth row reads: the square root of two times m cubed times n squared. The second column entry reads: two. The third column entry reads: m, n.\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Constants<\/th>\r\n<th>Variables<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1. <em>x<\/em> + 5<\/td>\r\n<td>5<\/td>\r\n<td><em>x<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2. [latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{4}{3},\\pi [\/latex]<\/td>\r\n<td>[latex]r[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\r\n<td>2<\/td>\r\n<td>[latex]m,n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109667&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating an Algebraic Expression at Different Values<\/h3>\r\nEvaluate the expression [latex]2x - 7[\/latex] for each value for <em>x.<\/em>\r\n<ol>\r\n \t<li>[latex]x=0[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]x=-4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"421675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421675\"]\r\n<ol>\r\n \t<li>Substitute 0 for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(0\\right)-7 \\\\ &amp; =0-7 \\\\ &amp; =-7\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 1 for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(1\\right)-7 \\\\ &amp; =2-7 \\\\ &amp; =-5\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(\\frac{1}{2}\\right)-7 \\\\ &amp; =1-7 \\\\ &amp; =-6\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(-4\\right)-7 \\\\ &amp; =-8-7 \\\\ &amp; =-15\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div style=\"text-align: center;\"><\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1976&amp;theme=oea&amp;iframe_resize_id=mom6[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating Algebraic Expressions<\/h3>\r\nEvaluate each expression for the given values.\r\n<ol>\r\n \t<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\r\n \t<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"182854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854\"]\r\n<ol>\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &amp;=\\left(-5\\right)+5 \\\\ &amp;=0\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 10 for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} &amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ &amp; =\\frac{10}{20-1} \\\\ &amp; =\\frac{10}{19}\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 5 for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} &amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ &amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ &amp; =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b &amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ &amp; =11-8-8 \\\\ &amp; =-85\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} &amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ &amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ &amp; =\\sqrt{144} \\\\ &amp; =12\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Be Careful when simplifying fractions!<\/h3>\r\nWhy does the fraction [latex]\\dfrac{(25)}{3(25)-1}[\/latex] not simplify to [latex]\\dfrac{\\cancel{(25)}}{3\\cancel{(25)}-1}=\\dfrac{1}{3-1}=\\dfrac{1}{2}[\/latex]?\r\n\r\nUsing the inverse property of multiplication, we are permitted to \"cancel out\" common factors in the numerator and denominator such that\u00a0[latex]\\dfrac{a}{a}=1[\/latex].\r\n\r\nBut be careful! We have no rule that allows us to cancel numbers in the top and bottom of a fractions that are contained in sums or differences. You'll see this idea reappear frequently throughout the course.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=483&amp;theme=oea&amp;iframe_resize_id=mom10[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92388&amp;theme=oea&amp;iframe_resize_id=mom12[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109700&amp;theme=oea&amp;iframe_resize_id=mom13[\/embed]\r\n\r\n<\/div>\r\nIn the following video we present more examples of how to evaluate an expression for a given value.\r\n\r\nhttps:\/\/youtu.be\/MkRdwV4n91g\r\n<h2>Formulas<\/h2>\r\nAn <strong>equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].\r\n\r\nA <strong>formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Formula<\/h3>\r\nA right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\\pi[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223817\/CNX_CAT_Figure_01_01_004.jpg\" alt=\"A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.\" width=\"487\" height=\"279\" \/> Right circular cylinder[\/caption]\r\n\r\n[reveal-answer q=\"257174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"257174\"]\r\n\r\nEvaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}S&amp;=2\\pi r\\left(r+h\\right) \\\\ &amp; =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ &amp; =2\\pi\\left(6\\right)\\left(15\\right) \\\\ &amp; =180\\pi\\end{align}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nThe surface area is [latex]180\\pi [\/latex] square inches.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<iframe src=\"https:\/\/lumenlearning.h5p.com\/content\/1290624917813565478\/embed\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"\/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/> <b>Figure 4<\/b>[\/caption]\r\n\r\nA photograph with length <em>L<\/em> and width <em>W<\/em> is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length 32 cm and width 24 cm.\r\n\r\n[reveal-answer q=\"846181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"846181\"]\r\n\r\n1,152 cm<sup>2<\/sup>\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]35044[\/ohm_question]\r\n\r\n[ohm_question]230571[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Simplify Algebraic Expressions<\/h2>\r\nSometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: operations on Fractions<\/h3>\r\nWhen simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.\r\n\r\n<strong>To multiply fractions<\/strong>, multiply the numerators and place them over the product of the denominators.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/p>\r\n<strong>To divide fractions<\/strong>, multiply the first by the reciprocal of the second.\r\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\r\n<strong>To simplify fractions<\/strong>, find common factors in the numerator and denominator that cancel.\r\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{24}{32}=\\dfrac{2\\cdot2\\cdot2\\cdot3}{2\\cdot2\\cdot2\\cdot2\\cdot2}=\\dfrac{3}{2\\cdot2}=\\dfrac{3}{4}[\/latex]<\/p>\r\n<strong>To add or subtract fractions<\/strong>, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.\r\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Algebraic Expressions<\/h3>\r\nSimplify each algebraic expression.\r\n<ol>\r\n \t<li>[latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\r\n \t<li>[latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\r\n \t<li>[latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\r\n \t<li>[latex]2mn - 5m+3mn+n[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"286046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286046\"]\r\n<ol>\r\n \t<li>[latex]\\begin{align}3x-2y+x-3y-7 &amp; =3x+x-2y-3y-7 &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =4x-5y-7 &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}2r-5\\left(3-r\\right)+4 &amp; =2r-15+5r+4 &amp;&amp; \\text{Distributive property}\\\\&amp;=2r+5r-15+4 &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =7r-11 &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &amp;=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &amp;&amp;\\text{Distributive property}\\\\&amp;=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s &amp;&amp; \\text{Commutative property of addition}\\\\&amp;=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s &amp;&amp; \\text{Common Denominators}\\\\ &amp; =\\frac{10}{3}t-\\frac{13}{4}s &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}mn-5m+3mn+n &amp; =2mn+3mn-5m+n &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =5mn-5m+n &amp;&amp; \\text{Simplify}\\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=50617&amp;theme=oea&amp;iframe_resize_id=mom30[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1980&amp;theme=oea&amp;iframe_resize_id=mom40[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3616&amp;theme=oea&amp;iframe_resize_id=mom50[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying a Formula<\/h3>\r\nA rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.\r\n\r\n[reveal-answer q=\"921194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"921194\"]\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;P=L+W+L+W \\\\ &amp;P=L+L+W+W &amp;&amp; \\text{Commutative property of addition} \\\\ &amp;P=2L+2W &amp;&amp; \\text{Simplify} \\\\ &amp;P=2\\left(L+W\\right) &amp;&amp; \\text{Distributive property}\\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>List the constants and variables in an algebraic expression.<\/li>\n<li>Evaluate an algebraic expression.<\/li>\n<li>Use an algebraic formula.<\/li>\n<\/ul>\n<\/div>\n<p>So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as [latex]x+5,\\frac{4}{3}\\pi {r}^{3}[\/latex], or [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]. In the expression [latex]x+5, 5[\/latex] is called a <strong>constant<\/strong> because it does not vary and <em>x<\/em> is called a <strong>variable<\/strong> because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.<\/p>\n<p>We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}&\\left(-3\\right)^{5}=\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right)\\cdot\\left(-3\\right) && x^{5}=x\\cdot x\\cdot x\\cdot x\\cdot x \\\\ &\\left(2\\cdot7\\right)^{3}=\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right)\\cdot\\left(2\\cdot7\\right) && \\left(yz\\right)^{3}=\\left(yz\\right)\\cdot\\left(yz\\right)\\cdot\\left(yz\\right)\\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<p>In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.<\/p>\n<p>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Describing Algebraic Expressions<\/h3>\n<p>List the constants and variables for each algebraic expression.<\/p>\n<ol>\n<li><em>x<\/em> + 5<\/li>\n<li>[latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q790423\">Show Solution<\/span><\/p>\n<div id=\"q790423\" class=\"hidden-answer\" style=\"display: none\">\n<table summary=\"A table with four rows and three columns. The first entry of the first row is empty, but the second entry reads: Constants, and the third reads: Variables. The first entry of the second row reads: x plus five. The second column entry reads: five. The third column entry reads: x. The first entry of the third row reads: four-thirds pi times r cubed. The second column entry reads: four-thirds, pi. The third column entry reads: r. The first entry of the fourth row reads: the square root of two times m cubed times n squared. The second column entry reads: two. The third column entry reads: m, n.\">\n<thead>\n<tr>\n<th><\/th>\n<th>Constants<\/th>\n<th>Variables<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1. <em>x<\/em> + 5<\/td>\n<td>5<\/td>\n<td><em>x<\/em><\/td>\n<\/tr>\n<tr>\n<td>2. [latex]\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/td>\n<td>[latex]\\frac{4}{3},\\pi[\/latex]<\/td>\n<td>[latex]r[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3. [latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex]<\/td>\n<td>2<\/td>\n<td>[latex]m,n[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109667\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109667&#38;theme=oea&#38;iframe_resize_id=ohm109667&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating an Algebraic Expression at Different Values<\/h3>\n<p>Evaluate the expression [latex]2x - 7[\/latex] for each value for <em>x.<\/em><\/p>\n<ol>\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]x=-4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q421675\">Show Solution<\/span><\/p>\n<div id=\"q421675\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute 0 for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(0\\right)-7 \\\\ & =0-7 \\\\ & =-7\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 1 for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(1\\right)-7 \\\\ & =2-7 \\\\ & =-5\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(\\frac{1}{2}\\right)-7 \\\\ & =1-7 \\\\ & =-6\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(-4\\right)-7 \\\\ & =-8-7 \\\\ & =-15\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1976\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1976&#38;theme=oea&#38;iframe_resize_id=ohm1976&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Algebraic Expressions<\/h3>\n<p>Evaluate each expression for the given values.<\/p>\n<ol>\n<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\n<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\n<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182854\">Show Solution<\/span><\/p>\n<div id=\"q182854\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &=\\left(-5\\right)+5 \\\\ &=0\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 10 for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} & =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ & =\\frac{10}{20-1} \\\\ & =\\frac{10}{19}\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 5 for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} & =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ & =\\frac{4}{3}\\pi\\left(125\\right) \\\\ & =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 11 for [latex]a[\/latex] and \u20138 for [latex]b[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b & =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ & =11-8-8 \\\\ & =-85\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 2 for [latex]m[\/latex] and 3 for [latex]n[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} & =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ & =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ & =\\sqrt{144} \\\\ & =12\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Be Careful when simplifying fractions!<\/h3>\n<p>Why does the fraction [latex]\\dfrac{(25)}{3(25)-1}[\/latex] not simplify to [latex]\\dfrac{\\cancel{(25)}}{3\\cancel{(25)}-1}=\\dfrac{1}{3-1}=\\dfrac{1}{2}[\/latex]?<\/p>\n<p>Using the inverse property of multiplication, we are permitted to &#8220;cancel out&#8221; common factors in the numerator and denominator such that\u00a0[latex]\\dfrac{a}{a}=1[\/latex].<\/p>\n<p>But be careful! We have no rule that allows us to cancel numbers in the top and bottom of a fractions that are contained in sums or differences. You&#8217;ll see this idea reappear frequently throughout the course.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm483\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=483&#38;theme=oea&#38;iframe_resize_id=ohm483&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm92388\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92388&#38;theme=oea&#38;iframe_resize_id=ohm92388&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm109700\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109700&#38;theme=oea&#38;iframe_resize_id=ohm109700&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we present more examples of how to evaluate an expression for a given value.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Evaluate Various Algebraic Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Formulas<\/h2>\n<p>An <strong>equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute 3 for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].<\/p>\n<p>A <strong>formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Formula<\/h3>\n<p>A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex]. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of [latex]\\pi[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223817\/CNX_CAT_Figure_01_01_004.jpg\" alt=\"A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.\" width=\"487\" height=\"279\" \/><\/p>\n<p class=\"wp-caption-text\">Right circular cylinder<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q257174\">Show Solution<\/span><\/p>\n<div id=\"q257174\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}S&=2\\pi r\\left(r+h\\right) \\\\ & =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ & =2\\pi\\left(6\\right)\\left(15\\right) \\\\ & =180\\pi\\end{align}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>The surface area is [latex]180\\pi[\/latex] square inches.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumenlearning.h5p.com\/content\/1290624917813565478\/embed\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"\/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p>A photograph with length <em>L<\/em> and width <em>W<\/em> is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length 32 cm and width 24 cm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q846181\">Show Solution<\/span><\/p>\n<div id=\"q846181\" class=\"hidden-answer\" style=\"display: none\">\n<p>1,152 cm<sup>2<\/sup><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm35044\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35044&theme=oea&iframe_resize_id=ohm35044&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm230571\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=230571&theme=oea&iframe_resize_id=ohm230571&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Simplify Algebraic Expressions<\/h2>\n<p>Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: operations on Fractions<\/h3>\n<p>When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.<\/p>\n<p><strong>To multiply fractions<\/strong>, multiply the numerators and place them over the product of the denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/p>\n<p><strong>To divide fractions<\/strong>, multiply the first by the reciprocal of the second.<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\n<p><strong>To simplify fractions<\/strong>, find common factors in the numerator and denominator that cancel.<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{24}{32}=\\dfrac{2\\cdot2\\cdot2\\cdot3}{2\\cdot2\\cdot2\\cdot2\\cdot2}=\\dfrac{3}{2\\cdot2}=\\dfrac{3}{4}[\/latex]<\/p>\n<p><strong>To add or subtract fractions<\/strong>, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Algebraic Expressions<\/h3>\n<p>Simplify each algebraic expression.<\/p>\n<ol>\n<li>[latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\n<li>[latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\n<li>[latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\n<li>[latex]2mn - 5m+3mn+n[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q286046\">Show Solution<\/span><\/p>\n<div id=\"q286046\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{align}3x-2y+x-3y-7 & =3x+x-2y-3y-7 && \\text{Commutative property of addition} \\\\ & =4x-5y-7 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}2r-5\\left(3-r\\right)+4 & =2r-15+5r+4 && \\text{Distributive property}\\\\&=2r+5r-15+4 && \\text{Commutative property of addition} \\\\ & =7r-11 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &&\\text{Distributive property}\\\\&=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s && \\text{Commutative property of addition}\\\\&=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s && \\text{Common Denominators}\\\\ & =\\frac{10}{3}t-\\frac{13}{4}s && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}mn-5m+3mn+n & =2mn+3mn-5m+n && \\text{Commutative property of addition} \\\\ & =5mn-5m+n && \\text{Simplify}\\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm50617\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=50617&#38;theme=oea&#38;iframe_resize_id=ohm50617&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm1980\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1980&#38;theme=oea&#38;iframe_resize_id=ohm1980&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm3616\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3616&#38;theme=oea&#38;iframe_resize_id=ohm3616&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying a Formula<\/h3>\n<p>A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q921194\">Show Solution<\/span><\/p>\n<div id=\"q921194\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{align}&P=L+W+L+W \\\\ &P=L+L+W+W && \\text{Commutative property of addition} \\\\ &P=2L+2W && \\text{Simplify} \\\\ &P=2\\left(L+W\\right) && \\text{Distributive property}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-41\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Area of a Rectangle Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/eq1cow0lcj\">https:\/\/www.desmos.com\/calculator\/eq1cow0lcj<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Quiestion ID 3616. <strong>Authored by<\/strong>: triplett,shawn. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1976, 1980. <strong>Authored by<\/strong>: Morales, Lawrence. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 483. <strong>Authored by<\/strong>: Eldridge,Jeff, mb Sousa,James. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 92388. <strong>Authored by<\/strong>: Jenck,Michael for Lumen Learning. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 109700, 110263, 109667. <strong>Authored by<\/strong>: Day, Alyson. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 50617. <strong>Authored by<\/strong>: Gardner,Brenda. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>:  IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Evaluating Algebraic Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/MkRdwV4n91g\">https:\/\/youtu.be\/MkRdwV4n91g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul 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