{"id":2803,"date":"2024-02-09T19:12:32","date_gmt":"2024-02-09T19:12:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2803"},"modified":"2024-02-09T19:12:35","modified_gmt":"2024-02-09T19:12:35","slug":"3-1-exponential-properties-for-multiplication-and-powers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/3-1-exponential-properties-for-multiplication-and-powers\/","title":{"raw":"3.1 Exponential Properties for Multiplication and Powers","rendered":"3.1 Exponential Properties for Multiplication and Powers"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions using the Product Property of Exponents<\/li>\r\n \t<li>Simplify expressions using the Power Property of Exponents<\/li>\r\n \t<li>Simplify expressions using the Product to a Power Property of Exponents<\/li>\r\n \t<li>Simplify expressions using the Quotient Property of Exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Product<\/strong>: the result when two or more numbers are multiplied<\/li>\r\n \t<li><strong>Power<\/strong>: exponent<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Properties of Exponents<\/h2>\r\nExponents have certain properties that arise due to the properties of multiplication. We\u2019ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.\u00a0 First, we will look at a few examples that leads to the Product Property.\r\n\r\nFor example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex]. The exponent only applies to the number immediately to its left, unless there are parentheses. So,\u00a0[latex]-5^{4}=-(5\\cdot5\\cdot5\\cdot5)=-625[\/latex] and\u00a0[latex](-5)^{4}=(-5\\cdot -5\\cdot -5\\cdot -5)=625[\/latex].\r\n\r\nWhat happens if we multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each term, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2=2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents, [latex]3[\/latex] and [latex]4[\/latex].\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">The base stayed the same and we added the exponents. This is an example of the <em><strong>Product Property for Exponents<\/strong><\/em>.<\/span>\r\n<div class=\"textbox shaded\">\r\n<h3>The Product Property OF Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and any integers [latex]m[\/latex]\u00a0and [latex]n[\/latex], \u00a0[latex]a^{m}\\cdot a^{n} = a^{m+n}[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo multiply exponential terms with the same base, add the exponents.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When we are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product property we may only apply it when the terms being multiplied have the<span style=\"color: #0000ff;\"> same base<\/span> and the <span style=\"color: #0000ff;\">exponents are integers<\/span>. Conditions on mathematical rules are often given before the rule is stated, as in this example it says \"For any real number [latex]a[\/latex] and any integers [latex]m[\/latex]\u00a0and [latex]n[\/latex].\"\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{2}^{5}\\cdot {2}^{7}[\/latex]\r\n\r\n[reveal-answer q=\"RB010\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"RB010\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466105371\" class=\"unnumbered unstyled\" summary=\"The top line says x to the 5th times x to the 7th. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{2}^{5}\\cdot {2}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n<td>[latex]2^{\\color{red}{5+7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\left (\\frac{1}{4}\\right )^{3}\\left (\\frac{1}{4}\\right )^{7}[\/latex]<\/p>\r\n[reveal-answer q=\"356596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356596\"]\r\n\r\nThe base of both exponents is [latex]\\frac{1}{4}[\/latex], so the product rule applies.\r\n<p style=\"text-align: center;\">[latex]\\left (\\frac{1}{4}\\right )^{3}\\left (\\frac{1}{4}\\right )^{7}[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center;\">\u00a0 = [latex]\\left (\\frac{1}{4}\\right )^{3+7}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">= [latex]\\left (\\frac{1}{4}\\right )^{10}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left ( {\\frac{1}{2}}\\right )^{7}\\cdot \\left ( {\\frac{1}{2}}\\right )^{9}[\/latex]\r\n[reveal-answer q=\"971008\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971008\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469803212\" class=\"unnumbered unstyled\" summary=\"The top line says 2 to the 7th times 2 to the 9th. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left ( {\\frac{1}{2}}\\right )^{7}\\cdot \\left ( {\\frac{1}{2}}\\right )^{9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n<td>[latex]\\left ( \\frac{1}{2}\\right )^{\\color{red}{7+9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\left ({\\frac{1}{2}}\\right )^{16}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146143[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659.png\" alt=\"traffic-sign-160659\" width=\"96\" height=\"83\" \/><\/h3>\r\nCaution! Do not try to apply this rule to sums.\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\nThink about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]\r\n<p style=\"text-align: center;\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\r\nNo, it does not because of the order of operations!\r\n<p style=\"text-align: center;\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\r\n<p style=\"text-align: center;\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\r\nTherefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).\r\n\r\n<\/div>\r\n<h3>Simplify Expressions Using the Power Property of Exponents<\/h3>\r\nWe will now further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power.\u00a0\u00a0Let's see if we can discover a general property.\r\n\r\nLet\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is [latex]4[\/latex], so we multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).\r\n\r\n[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of [latex]5[\/latex] to the second power, and we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].\r\n\r\nSo, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if we do the multiplication).\r\n\r\nThis leads to another rule for exponents\u2014the <em><b>Power Property for Exponents<\/b><\/em>. To simplify a power of a power, we multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{3\\cdot 5}=2^{15}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Power Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] is a real number and [latex]m,n[\/latex] are whole numbers, then\u00a0[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo raise a power to a power, multiply the exponents.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nAn example with numbers helps to verify this property.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left({5}^{2}\\right)}^{3}&amp; \\stackrel{?}{=}&amp; {5}^{2\\cdot 3}\\hfill \\\\ \\hfill {\\left(25\\right)}^{3}&amp; \\stackrel{?}{=}&amp; {5}^{6}\\hfill \\\\ \\hfill 15,625&amp; =&amp; 15,625\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left({2}^{5}\\right)}^{7}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\left [ {\\left(\\frac{1}{{3}}\\right)^{6}}\\right ] ^{8}[\/latex]\r\n\r\n[reveal-answer q=\"411160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411160\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468674718\" class=\"unnumbered unstyled\" summary=\"The top line shows x to the 5th in parentheses raised to the 7th. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left({2}^{5}\\right)}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\r\n<td>[latex]2^{\\color{red}{5\\cdot{7}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{35}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168047388004\" class=\"unnumbered unstyled\" summary=\"The top line shows 3 to the 6th in parentheses raised to the 8th power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left [ {\\left(\\frac{1}{{3}}\\right)^{6}}\\right ] ^{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\r\n<td>[latex]{\\left(\\frac{1}{3}\\right)}^{\\color{red}{6\\cdot{8}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\left(\\frac{1}{3}\\right)}^{48}[\/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 tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\left ((-3)^{5}\\right )^{8}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\left [ \\left (\\frac{3}{4}\\right )^{6}\\right ]^{9}[\/latex]\r\n\r\n[reveal-answer q=\"964056\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"964056\"]\r\n\r\n&nbsp;\r\n\r\n1. [latex](-3)^{40}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\left (\\frac{3}{4}\\right )^{54}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(-3\\right)}^{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2 [latex]-{3}^{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\nSolution\r\n<table id=\"eip-id1168465147148\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 268.84375px;\">1.<\/td>\r\n<td style=\"width: 160.390625px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 268.84375px;\">The exponent applies to the base, [latex]-3[\/latex] .<\/td>\r\n<td style=\"width: 160.390625px;\">[latex]{\\left(-3\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 268.84375px;\">Simplify.<\/td>\r\n<td style=\"width: 160.390625px;\">[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466253794\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The expression [latex]-{3}^{2}[\/latex] means: find the opposite of [latex]{3}^{2}[\/latex]\r\n\r\nThe exponent applies only to the base, [latex]3[\/latex].<\/td>\r\n<td>[latex]-{3}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as a product with [latex]\u22121[\/latex].<\/td>\r\n<td>[latex]-1\\cdot {3}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the reciprocal of the base and change the sign of the exponent.<\/td>\r\n<td>[latex]-1\\cdot9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(-2\\right)}^{4}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2 [latex]-{2}^{4}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"575008\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"575008\"]\r\n\r\n1. [latex]{\\left(-2\\right)}^{4}=16[\/latex]\r\n\r\n&nbsp;\r\n\r\n2 [latex]-{2}^{4}= -16[\/latex]\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We\u2019ll see how this works in the next example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]5\\cdot {2}^{3}[\/latex]\r\n\r\n[reveal-answer q=\"633196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"633196\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469497194\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice the exponent applies to just the base [latex]2[\/latex] .<\/td>\r\n<td>[latex]5\\cdot {2}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Evaluate the reciprocal.<\/td>\r\n<td>[latex]5\\cdot 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]40[\/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\nSimplify:\u00a0 [latex]-3\\cdot {4}^{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"425900\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"425900\"]\r\n\r\n[latex]-3\\cdot {4}^{2}=-48[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\nIn the next example, examples 1 and 2 look similar, but we get different results.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{-2}^{4}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]{\\left(-2\\right)}^{4}[\/latex]\r\n\r\n&nbsp;\r\n<h4>Solution<\/h4>\r\nRemember to always follow the order of operations.\r\n\r\n1. [latex]{-2}^{4}[\/latex]\r\n\r\n[latex]= -16[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]{\\left(-2\\right)}^{4}[\/latex]\r\n\r\n[latex]=16[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{-3}^{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]{\\left(-5\\right)}^{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"758294\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"758294\"]\r\n\r\n1. [latex]{-3}^{2}=-9[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]{\\left(-5\\right)}^{2}=25[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Simplify Expressions Using the Product to a Power Property<\/h3>\r\nWe will now look at an expression containing a product that is raised to a power. Look for a pattern.\r\n\r\nSimplify this expression.\r\n<p style=\"text-align: left;\">\u00a0 \u00a0[latex]\\left(2\\cdot3\\right)^{4}[\/latex] \u00a0 \u00a0 Base =\u00a0[latex]\\left(2\\cdot3\\right)[\/latex]; exponent =\u00a0[latex]4[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]=\\left(2\\cdot 3\\right)\\left(2\\cdot 3\\right)\\left(2\\cdot 3\\right)\\left(2\\cdot 3\\right)[\/latex] \u00a0 \u00a0 \u00a0The base gets multiplied 4 times.<\/p>\r\n<p style=\"text-align: left;\">[latex]=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(3\\cdot{3}\\cdot{3}\\cdot{3}\\right)[\/latex] \u00a0 \u00a0 Regroup using the commutative and associative properties of multiplication.<\/p>\r\n<p style=\"text-align: left;\">[latex]=\\left(2^{4}\\right)\\left(3^{4}\\right)[\/latex] \u00a0 \u00a0 Rewrite using exponential notation.<\/p>\r\n<p style=\"text-align: left;\">[latex]=16\\cdot 81[\/latex]<\/p>\r\nNotice that the exponent is applied to each factor of [latex]2\\cdot 3[\/latex]. So, we can eliminate the middle steps.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2\\cdot 3\\right)^{4} = \\left(2^{4}\\right)\\left(3^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }3\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\;\\;\\;\\;=16\\cdot81\\end{array}[\/latex]<\/p>\r\nThe product of two or more numbers raised to a power is equal to the product of each number raised to the same power.\r\n\r\nThe exponent applies to each of the factors. This leads to the <em><strong>Product to a Power Property for Exponents<\/strong><\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3>Product to a Power Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]n[\/latex] is a whole number, then\u00a0[latex]{\\left(a\\,b\\right)}^{n}={a}^{n}{b}^{n}[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo raise a product to a power, raise each factor to that power.\r\n\r\n<\/div>\r\nAn example with numbers helps to verify this property:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left(2\\cdot 3\\right)}^{2}&amp; \\stackrel{?}{=}&amp; {2}^{2}\\cdot {3}^{2}\\hfill \\\\ \\hfill {6}^{2}&amp; \\stackrel{?}{=}&amp; 4\\cdot 9\\hfill \\\\ \\hfill 36&amp; =&amp; 36\\hfill \\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(-11\\cdot 2\\right)}^{2}[\/latex]\r\n[reveal-answer q=\"390160\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390160\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466596049\" class=\"unnumbered unstyled\" summary=\"The top line shows negative 11x in parentheses, squared. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(-11\\cdot 2\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power of a Product Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\r\n<td>[latex](-11)^{\\color{red}{2}}2^{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]121\\cdot4=484[\/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 tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\u00a0[latex]{\\left(-3\\cdot 2\\right)}^{3}[\/latex]\r\n\r\n[reveal-answer q=\"590354\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"590354\"]\r\n<p style=\"text-align: left;\">[latex](-3)^{3}\\cdot (2)^{3}=-27\\cdot 8=-216[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWe have developed the properties of exponents for multiplication:\r\n<div class=\"textbox shaded\">\r\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]m[\/latex] and [latex]n[\/latex] are whole numbers, then,\r\n\r\nProduct Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]\r\n\r\nPower Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\left (a^m\\right )^n=a^{m\\cdot n}[\/latex]\r\n\r\nProduct to a Power: \u00a0 \u00a0 \u00a0[latex]\\left (ab\\right )^{n}={a}^{n}{b}^{n}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions using the Product Property of Exponents<\/li>\n<li>Simplify expressions using the Power Property of Exponents<\/li>\n<li>Simplify expressions using the Product to a Power Property of Exponents<\/li>\n<li>Simplify expressions using the Quotient Property of Exponents<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Product<\/strong>: the result when two or more numbers are multiplied<\/li>\n<li><strong>Power<\/strong>: exponent<\/li>\n<\/ul>\n<\/div>\n<h2>Properties of Exponents<\/h2>\n<p>Exponents have certain properties that arise due to the properties of multiplication. We\u2019ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.\u00a0 First, we will look at a few examples that leads to the Product Property.<\/p>\n<p>For example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or [latex]625[\/latex]. The exponent only applies to the number immediately to its left, unless there are parentheses. So,\u00a0[latex]-5^{4}=-(5\\cdot5\\cdot5\\cdot5)=-625[\/latex] and\u00a0[latex](-5)^{4}=(-5\\cdot -5\\cdot -5\\cdot -5)=625[\/latex].<\/p>\n<p>What happens if we multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each term, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2=2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents, [latex]3[\/latex] and [latex]4[\/latex].<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">The base stayed the same and we added the exponents. This is an example of the <em><strong>Product Property for Exponents<\/strong><\/em>.<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>The Product Property OF Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and any integers [latex]m[\/latex]\u00a0and [latex]n[\/latex], \u00a0[latex]a^{m}\\cdot a^{n} = a^{m+n}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To multiply exponential terms with the same base, add the exponents.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When we are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product property we may only apply it when the terms being multiplied have the<span style=\"color: #0000ff;\"> same base<\/span> and the <span style=\"color: #0000ff;\">exponents are integers<\/span>. Conditions on mathematical rules are often given before the rule is stated, as in this example it says &#8220;For any real number [latex]a[\/latex] and any integers [latex]m[\/latex]\u00a0and [latex]n[\/latex].&#8221;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{2}^{5}\\cdot {2}^{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qRB010\">Show Solution<\/span><\/p>\n<div id=\"qRB010\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466105371\" class=\"unnumbered unstyled\" summary=\"The top line says x to the 5th times x to the 7th. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{2}^{5}\\cdot {2}^{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\n<td>[latex]2^{\\color{red}{5+7}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\left (\\frac{1}{4}\\right )^{3}\\left (\\frac{1}{4}\\right )^{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q356596\">Show Solution<\/span><\/p>\n<div id=\"q356596\" class=\"hidden-answer\" style=\"display: none\">\n<p>The base of both exponents is [latex]\\frac{1}{4}[\/latex], so the product rule applies.<\/p>\n<p style=\"text-align: center;\">[latex]\\left (\\frac{1}{4}\\right )^{3}\\left (\\frac{1}{4}\\right )^{7}[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">\u00a0 = [latex]\\left (\\frac{1}{4}\\right )^{3+7}[\/latex]<\/p>\n<p style=\"text-align: center;\">= [latex]\\left (\\frac{1}{4}\\right )^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left ( {\\frac{1}{2}}\\right )^{7}\\cdot \\left ( {\\frac{1}{2}}\\right )^{9}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q971008\">Show Solution<\/span><\/p>\n<div id=\"q971008\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469803212\" class=\"unnumbered unstyled\" summary=\"The top line says 2 to the 7th times 2 to the 9th. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left ( {\\frac{1}{2}}\\right )^{7}\\cdot \\left ( {\\frac{1}{2}}\\right )^{9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the product property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\n<td>[latex]\\left ( \\frac{1}{2}\\right )^{\\color{red}{7+9}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\left ({\\frac{1}{2}}\\right )^{16}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146143\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146143&theme=oea&iframe_resize_id=ohm146143&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659.png\" alt=\"traffic-sign-160659\" width=\"96\" height=\"83\" \/><\/h3>\n<p>Caution! Do not try to apply this rule to sums.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Think about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\n<p>No, it does not because of the order of operations!<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\n<p style=\"text-align: center;\">and<\/p>\n<p style=\"text-align: center;\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\n<p>Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).<\/p>\n<\/div>\n<h3>Simplify Expressions Using the Power Property of Exponents<\/h3>\n<p>We will now further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power.\u00a0\u00a0Let&#8217;s see if we can discover a general property.<\/p>\n<p>Let\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is [latex]4[\/latex], so we multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).<\/p>\n<p>[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of [latex]5[\/latex] to the second power, and we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].<\/p>\n<p>So, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if we do the multiplication).<\/p>\n<p>This leads to another rule for exponents\u2014the <em><b>Power Property for Exponents<\/b><\/em>. To simplify a power of a power, we multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{3\\cdot 5}=2^{15}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Power Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] is a real number and [latex]m,n[\/latex] are whole numbers, then\u00a0[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To raise a power to a power, multiply the exponents.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>An example with numbers helps to verify this property.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left({5}^{2}\\right)}^{3}& \\stackrel{?}{=}& {5}^{2\\cdot 3}\\hfill \\\\ \\hfill {\\left(25\\right)}^{3}& \\stackrel{?}{=}& {5}^{6}\\hfill \\\\ \\hfill 15,625& =& 15,625\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left({2}^{5}\\right)}^{7}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\left [ {\\left(\\frac{1}{{3}}\\right)^{6}}\\right ] ^{8}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q411160\">Show Solution<\/span><\/p>\n<div id=\"q411160\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468674718\" class=\"unnumbered unstyled\" summary=\"The top line shows x to the 5th in parentheses raised to the 7th. The next line says,\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\left({2}^{5}\\right)}^{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\n<td>[latex]2^{\\color{red}{5\\cdot{7}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{35}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168047388004\" class=\"unnumbered unstyled\" summary=\"The top line shows 3 to the 6th in parentheses raised to the 8th power. The next line says,\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\left [ {\\left(\\frac{1}{{3}}\\right)^{6}}\\right ] ^{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\n<td>[latex]{\\left(\\frac{1}{3}\\right)}^{\\color{red}{6\\cdot{8}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\left(\\frac{1}{3}\\right)}^{48}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\left ((-3)^{5}\\right )^{8}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\left [ \\left (\\frac{3}{4}\\right )^{6}\\right ]^{9}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q964056\">Show Answer<\/span><\/p>\n<div id=\"q964056\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>1. [latex](-3)^{40}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\left (\\frac{3}{4}\\right )^{54}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(-3\\right)}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2 [latex]-{3}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168465147148\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 268.84375px;\">1.<\/td>\n<td style=\"width: 160.390625px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 268.84375px;\">The exponent applies to the base, [latex]-3[\/latex] .<\/td>\n<td style=\"width: 160.390625px;\">[latex]{\\left(-3\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 268.84375px;\">Simplify.<\/td>\n<td style=\"width: 160.390625px;\">[latex]9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466253794\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The expression [latex]-{3}^{2}[\/latex] means: find the opposite of [latex]{3}^{2}[\/latex]<\/p>\n<p>The exponent applies only to the base, [latex]3[\/latex].<\/td>\n<td>[latex]-{3}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as a product with [latex]\u22121[\/latex].<\/td>\n<td>[latex]-1\\cdot {3}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the reciprocal of the base and change the sign of the exponent.<\/td>\n<td>[latex]-1\\cdot9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(-2\\right)}^{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2 [latex]-{2}^{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q575008\">Show Answer<\/span><\/p>\n<div id=\"q575008\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]{\\left(-2\\right)}^{4}=16[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2 [latex]-{2}^{4}= -16[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We\u2019ll see how this works in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]5\\cdot {2}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q633196\">Show Solution<\/span><\/p>\n<div id=\"q633196\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469497194\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice the exponent applies to just the base [latex]2[\/latex] .<\/td>\n<td>[latex]5\\cdot {2}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Evaluate the reciprocal.<\/td>\n<td>[latex]5\\cdot 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]40[\/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>Simplify:\u00a0 [latex]-3\\cdot {4}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q425900\">Show Answer<\/span><\/p>\n<div id=\"q425900\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-3\\cdot {4}^{2}=-48[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<p>In the next example, examples 1 and 2 look similar, but we get different results.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{-2}^{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]{\\left(-2\\right)}^{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4>Solution<\/h4>\n<p>Remember to always follow the order of operations.<\/p>\n<p>1. [latex]{-2}^{4}[\/latex]<\/p>\n<p>[latex]= -16[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]{\\left(-2\\right)}^{4}[\/latex]<\/p>\n<p>[latex]=16[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{-3}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]{\\left(-5\\right)}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q758294\">Show Answer<\/span><\/p>\n<div id=\"q758294\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]{-3}^{2}=-9[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]{\\left(-5\\right)}^{2}=25[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Simplify Expressions Using the Product to a Power Property<\/h3>\n<p>We will now look at an expression containing a product that is raised to a power. Look for a pattern.<\/p>\n<p>Simplify this expression.<\/p>\n<p style=\"text-align: left;\">\u00a0 \u00a0[latex]\\left(2\\cdot3\\right)^{4}[\/latex] \u00a0 \u00a0 Base =\u00a0[latex]\\left(2\\cdot3\\right)[\/latex]; exponent =\u00a0[latex]4[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]=\\left(2\\cdot 3\\right)\\left(2\\cdot 3\\right)\\left(2\\cdot 3\\right)\\left(2\\cdot 3\\right)[\/latex] \u00a0 \u00a0 \u00a0The base gets multiplied 4 times.<\/p>\n<p style=\"text-align: left;\">[latex]=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(3\\cdot{3}\\cdot{3}\\cdot{3}\\right)[\/latex] \u00a0 \u00a0 Regroup using the commutative and associative properties of multiplication.<\/p>\n<p style=\"text-align: left;\">[latex]=\\left(2^{4}\\right)\\left(3^{4}\\right)[\/latex] \u00a0 \u00a0 Rewrite using exponential notation.<\/p>\n<p style=\"text-align: left;\">[latex]=16\\cdot 81[\/latex]<\/p>\n<p>Notice that the exponent is applied to each factor of [latex]2\\cdot 3[\/latex]. So, we can eliminate the middle steps.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2\\cdot 3\\right)^{4} = \\left(2^{4}\\right)\\left(3^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }3\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\;\\;\\;\\;=16\\cdot81\\end{array}[\/latex]<\/p>\n<p>The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.<\/p>\n<p>The exponent applies to each of the factors. This leads to the <em><strong>Product to a Power Property for Exponents<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Product to a Power Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]n[\/latex] is a whole number, then\u00a0[latex]{\\left(a\\,b\\right)}^{n}={a}^{n}{b}^{n}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To raise a product to a power, raise each factor to that power.<\/p>\n<\/div>\n<p>An example with numbers helps to verify this property:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left(2\\cdot 3\\right)}^{2}& \\stackrel{?}{=}& {2}^{2}\\cdot {3}^{2}\\hfill \\\\ \\hfill {6}^{2}& \\stackrel{?}{=}& 4\\cdot 9\\hfill \\\\ \\hfill 36& =& 36\\hfill \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(-11\\cdot 2\\right)}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q390160\">Show Solution<\/span><\/p>\n<div id=\"q390160\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466596049\" class=\"unnumbered unstyled\" summary=\"The top line shows negative 11x in parentheses, squared. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(-11\\cdot 2\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power of a Product Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\n<td>[latex](-11)^{\\color{red}{2}}2^{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]121\\cdot4=484[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:\u00a0[latex]{\\left(-3\\cdot 2\\right)}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q590354\">Show Answer<\/span><\/p>\n<div id=\"q590354\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex](-3)^{3}\\cdot (2)^{3}=-27\\cdot 8=-216[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>We have developed the properties of exponents for multiplication:<\/p>\n<div class=\"textbox shaded\">\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]m[\/latex] and [latex]n[\/latex] are whole numbers, then,<\/p>\n<p>Product Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/p>\n<p>Power Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\left (a^m\\right )^n=a^{m\\cdot n}[\/latex]<\/p>\n<p>Product to a Power: \u00a0 \u00a0 \u00a0[latex]\\left (ab\\right )^{n}={a}^{n}{b}^{n}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2><\/h2>\n","protected":false},"author":370291,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2803","chapter","type-chapter","status-publish","hentry"],"part":2801,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2803","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2803\/revisions"}],"predecessor-version":[{"id":2804,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2803\/revisions\/2804"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2801"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2803\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2803"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2803"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2803"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}