{"id":2212,"date":"2016-07-06T23:57:19","date_gmt":"2016-07-06T23:57:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=2212"},"modified":"2019-08-06T18:36:16","modified_gmt":"2019-08-06T18:36:16","slug":"read-find-the-power-of-a-product-and-a-quotient","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-find-the-power-of-a-product-and-a-quotient\/","title":{"raw":"Find the Power of a Product and a Quotient","rendered":"Find the Power of a Product and a Quotient"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify compound expressions using the exponent rules<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Finding the Power of a Product<\/h2>\r\nTo simplify the power of a product of two exponential expressions, we can use the <em>power of a product rule of exponents\u00a0<\/em>which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider [latex]{\\left(pq\\right)}^{3}[\/latex]. We begin by using the associative and commutative properties of multiplication to regroup the factors.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left(pq\\right)}^{3}&amp; =&amp; \\stackrel{3\\text{ factors}}{{\\left(pq\\right)\\cdot \\left(pq\\right)\\cdot \\left(pq\\right)}}\\hfill \\\\ &amp; =&amp; p\\cdot q\\cdot p\\cdot q\\cdot p\\cdot q\\hfill \\\\ &amp; =&amp; \\stackrel{3\\text{ factors}}{{p\\cdot p\\cdot p}}\\cdot \\stackrel{3\\text{ factors}}{{q\\cdot q\\cdot q}}\\hfill \\\\ &amp; =&amp; {p}^{3}\\cdot {q}^{3}\\hfill \\end{array}[\/latex]<\/div>\r\nIn other words, [latex]{\\left(pq\\right)}^{3}={p}^{3}\\cdot {q}^{3}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>The Power of a Product Rule of Exponents<\/h3>\r\nFor any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a product rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.\r\n<ol>\r\n \t<li>[latex]{\\left(a{b}^{2}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(2^a{t}\\right)}^{15}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{1}{{\\left(-7z\\right)}^{4}}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"788982\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"788982\"]\r\n\r\nUse the product and quotient rules and the new definitions to simplify each expression.\r\n<ol>\r\n \t<li>[latex]{\\left(a{b}^{2}\\right)}^{3}={\\left(a\\right)}^{3}\\cdot {\\left({b}^{2}\\right)}^{3}={a}^{1\\cdot 3}\\cdot {b}^{2\\cdot 3}={a}^{3}{b}^{6}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(2^a{t}\\right)}^{15}={\\left(2^a\\right)}^{15}\\cdot {\\left(t\\right)}^{15}={2}^{a\\cdot15}\\cdot{t}^{15}=2^{15a}\\cdot{t}^{15}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}={\\left(-2\\right)}^{3}\\cdot {\\left({w}^{3}\\right)}^{3}=-8\\cdot {w}^{3\\cdot 3}=-8{w}^{9}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{1}{{\\left(-7z\\right)}^{4}}=\\frac{1}{{\\left(-7\\right)}^{4}\\cdot {\\left(z\\right)}^{4}}=\\frac{1}{2,401{z}^{4}}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}={\\left({e}^{-2}\\right)}^{7}\\cdot {\\left({f}^{2}\\right)}^{7}={e}^{-2\\cdot 7}\\cdot {f}^{2\\cdot 7}={e}^{-14}{f}^{14}=\\frac{{f}^{14}}{{e}^{14}}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\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\/wp-content\/uploads\/sites\/121\/2016\/06\/01183308\/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\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.\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">In the following video, \u00a0we provide more examples of how to find the power of a product.<\/span>\r\n\r\nhttps:\/\/youtu.be\/p-2UkpJQWpo\r\n<h2>Finding the Power of a Quotient<\/h2>\r\nTo simplify the power of a quotient of two expressions, we can use the <em>power of a quotient rule\u00a0<\/em>which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, look at the following example:\r\n<div style=\"text-align: center;\">[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}=\\frac{{f}^{14}}{{e}^{14}}[\/latex]<\/div>\r\nRewrite the original problem differently and look at the result:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left({e}^{-2}{f}^{2}\\right)}^{7}&amp; =&amp; {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7}\\hfill \\\\ &amp; =&amp; \\frac{{f}^{14}}{{e}^{14}}\\hfill \\end{array}[\/latex]<\/div>\r\nIt appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.\r\n<div style=\"text-align: center;\">[latex]\\normalsize\\begin{array}{ccc}\\hfill {\\left({e}^{-2}{f}^{2}\\right)}^{7}&amp; =&amp; {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7}\\hfill \\\\ &amp; =&amp; \\frac{{\\left({f}^{2}\\right)}^{7}}{{\\left({e}^{2}\\right)}^{7}}\\hfill \\\\ &amp; =&amp; \\frac{{f}^{2\\cdot 7}}{{e}^{2\\cdot 7}}\\hfill \\\\ &amp; =&amp; \\frac{{f}^{14}}{{e}^{14}}\\hfill \\end{array}[\/latex]<\/div>\r\n<div style=\"text-align: left;\">\r\n<div class=\"textbox shaded\">\r\n<h3>The Power of a Quotient Rule of Exponents<\/h3>\r\nFor any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a quotient rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]{\\left(\\frac{a}{b}\\right)}^{n}=\\frac{{a}^{n}}{{b}^{n}}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.\r\n<ol>\r\n \t<li>[latex]{\\left(\\frac{4}{{z}^{11}}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\frac{p}{{q}^{3}}\\right)}^{6}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\frac{-1}{{t}^{2}}\\right)}^{27}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({j}^{3}{k}^{-2}\\right)}^{4}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({m}^{-2}{n}^{-2}\\right)}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"660878\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"660878\"]\r\n<ol>\r\n \t<li>[latex]\\Large{\\left(\\frac{4}{{z}^{11}}\\right)}^{3}=\\frac{{\\left(4\\right)}^{3}}{{\\left({z}^{11}\\right)}^{3}}=\\frac{64}{{z}^{11\\cdot 3}}=\\frac{64}{{z}^{33}}[\/latex]<\/li>\r\n \t<li>[latex]\\Large{\\left(\\frac{p}{{q}^{3}}\\right)}^{6}=\\frac{{\\left(p\\right)}^{6}}{{\\left({q}^{3}\\right)}^{6}}=\\frac{{p}^{1\\cdot 6}}{{q}^{3\\cdot 6}}=\\frac{{p}^{6}}{{q}^{18}}[\/latex]<\/li>\r\n \t<li>[latex]\\Large{\\left(\\frac{-1}{{t}^{2}}\\right)}^{27}=\\frac{{\\left(-1\\right)}^{27}}{{\\left({t}^{2}\\right)}^{27}}=\\frac{-1}{{t}^{2\\cdot 27}}=\\frac{-1}{{t}^{54}}=-\\frac{1}{{t}^{54}}[\/latex]<\/li>\r\n \t<li>[latex]\\Large{\\left({j}^{3}{k}^{-2}\\right)}^{4}={\\left(\\frac{{j}^{3}}{{k}^{2}}\\right)}^{4}=\\frac{{\\left({j}^{3}\\right)}^{4}}{{\\left({k}^{2}\\right)}^{4}}=\\frac{{j}^{3\\cdot 4}}{{k}^{2\\cdot 4}}=\\frac{{j}^{12}}{{k}^{8}}[\/latex]<\/li>\r\n \t<li>[latex]\\Large{\\left({m}^{-2}{n}^{-2}\\right)}^{3}={\\left(\\frac{1}{{m}^{2}{n}^{2}}\\right)}^{3}=\\frac{{\\left(1\\right)}^{3}}{{\\left({m}^{2}{n}^{2}\\right)}^{3}}=\\frac{1}{{\\left({m}^{2}\\right)}^{3}{\\left({n}^{2}\\right)}^{3}}=\\frac{1}{{m}^{2\\cdot 3}\\cdot {n}^{2\\cdot 3}}=\\frac{1}{{m}^{6}{n}^{6}}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">The following video provides more examples of simplifying expressions using the power of a quotient and other exponent rules.<\/span>\r\n\r\nhttps:\/\/youtu.be\/BoBe31pRxFM\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.<\/li>\r\n \t<li>The product rule for exponents:\u00a0For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/li>\r\n \t<li>The quotient rule for exponents:\u00a0For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/li>\r\n \t<li>The power rule for exponents:\r\n<ol>\r\n \t<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <em>n<\/em>, [latex]\\left(ab\\right)^{n}=a^{n}\\cdot{b^{n}}[\/latex].<\/li>\r\n \t<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <em>n<\/em>, [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{n}=\\frac{a^{n}}{b^{n}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify compound expressions using the exponent rules<\/li>\n<\/ul>\n<\/div>\n<h2>Finding the Power of a Product<\/h2>\n<p>To simplify the power of a product of two exponential expressions, we can use the <em>power of a product rule of exponents\u00a0<\/em>which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider [latex]{\\left(pq\\right)}^{3}[\/latex]. We begin by using the associative and commutative properties of multiplication to regroup the factors.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left(pq\\right)}^{3}& =& \\stackrel{3\\text{ factors}}{{\\left(pq\\right)\\cdot \\left(pq\\right)\\cdot \\left(pq\\right)}}\\hfill \\\\ & =& p\\cdot q\\cdot p\\cdot q\\cdot p\\cdot q\\hfill \\\\ & =& \\stackrel{3\\text{ factors}}{{p\\cdot p\\cdot p}}\\cdot \\stackrel{3\\text{ factors}}{{q\\cdot q\\cdot q}}\\hfill \\\\ & =& {p}^{3}\\cdot {q}^{3}\\hfill \\end{array}[\/latex]<\/div>\n<p>In other words, [latex]{\\left(pq\\right)}^{3}={p}^{3}\\cdot {q}^{3}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>The Power of a Product Rule of Exponents<\/h3>\n<p>For any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a product rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{\\left(a{b}^{2}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(2^a{t}\\right)}^{15}[\/latex]<\/li>\n<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]\\frac{1}{{\\left(-7z\\right)}^{4}}[\/latex]<\/li>\n<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q788982\">Show Solution<\/span><\/p>\n<div id=\"q788982\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the product and quotient rules and the new definitions to simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left(a{b}^{2}\\right)}^{3}={\\left(a\\right)}^{3}\\cdot {\\left({b}^{2}\\right)}^{3}={a}^{1\\cdot 3}\\cdot {b}^{2\\cdot 3}={a}^{3}{b}^{6}[\/latex]<\/li>\n<li>[latex]{\\left(2^a{t}\\right)}^{15}={\\left(2^a\\right)}^{15}\\cdot {\\left(t\\right)}^{15}={2}^{a\\cdot15}\\cdot{t}^{15}=2^{15a}\\cdot{t}^{15}[\/latex]<\/li>\n<li>[latex]{\\left(-2{w}^{3}\\right)}^{3}={\\left(-2\\right)}^{3}\\cdot {\\left({w}^{3}\\right)}^{3}=-8\\cdot {w}^{3\\cdot 3}=-8{w}^{9}[\/latex]<\/li>\n<li>[latex]\\frac{1}{{\\left(-7z\\right)}^{4}}=\\frac{1}{{\\left(-7\\right)}^{4}\\cdot {\\left(z\\right)}^{4}}=\\frac{1}{2,401{z}^{4}}[\/latex]<\/li>\n<li>[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}={\\left({e}^{-2}\\right)}^{7}\\cdot {\\left({f}^{2}\\right)}^{7}={e}^{-2\\cdot 7}\\cdot {f}^{2\\cdot 7}={e}^{-14}{f}^{14}=\\frac{{f}^{14}}{{e}^{14}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\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\/wp-content\/uploads\/sites\/121\/2016\/06\/01183308\/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>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.<\/p>\n<\/div>\n<p><span style=\"color: #000000;\">In the following video, \u00a0we provide more examples of how to find the power of a product.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Expressions Using Exponent Rules (Power of a Product)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/p-2UkpJQWpo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Finding the Power of a Quotient<\/h2>\n<p>To simplify the power of a quotient of two expressions, we can use the <em>power of a quotient rule\u00a0<\/em>which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, look at the following example:<\/p>\n<div style=\"text-align: center;\">[latex]{\\left({e}^{-2}{f}^{2}\\right)}^{7}=\\frac{{f}^{14}}{{e}^{14}}[\/latex]<\/div>\n<p>Rewrite the original problem differently and look at the result:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\left({e}^{-2}{f}^{2}\\right)}^{7}& =& {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7}\\hfill \\\\ & =& \\frac{{f}^{14}}{{e}^{14}}\\hfill \\end{array}[\/latex]<\/div>\n<p>It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.<\/p>\n<div style=\"text-align: center;\">[latex]\\normalsize\\begin{array}{ccc}\\hfill {\\left({e}^{-2}{f}^{2}\\right)}^{7}& =& {\\left(\\frac{{f}^{2}}{{e}^{2}}\\right)}^{7}\\hfill \\\\ & =& \\frac{{\\left({f}^{2}\\right)}^{7}}{{\\left({e}^{2}\\right)}^{7}}\\hfill \\\\ & =& \\frac{{f}^{2\\cdot 7}}{{e}^{2\\cdot 7}}\\hfill \\\\ & =& \\frac{{f}^{14}}{{e}^{14}}\\hfill \\end{array}[\/latex]<\/div>\n<div style=\"text-align: left;\">\n<div class=\"textbox shaded\">\n<h3>The Power of a Quotient Rule of Exponents<\/h3>\n<p>For any real numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]n[\/latex], the power of a quotient rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]{\\left(\\frac{a}{b}\\right)}^{n}=\\frac{{a}^{n}}{{b}^{n}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.<\/p>\n<ol>\n<li>[latex]{\\left(\\frac{4}{{z}^{11}}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(\\frac{p}{{q}^{3}}\\right)}^{6}[\/latex]<\/li>\n<li>[latex]{\\left(\\frac{-1}{{t}^{2}}\\right)}^{27}[\/latex]<\/li>\n<li>[latex]{\\left({j}^{3}{k}^{-2}\\right)}^{4}[\/latex]<\/li>\n<li>[latex]{\\left({m}^{-2}{n}^{-2}\\right)}^{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q660878\">Show Solution<\/span><\/p>\n<div id=\"q660878\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\Large{\\left(\\frac{4}{{z}^{11}}\\right)}^{3}=\\frac{{\\left(4\\right)}^{3}}{{\\left({z}^{11}\\right)}^{3}}=\\frac{64}{{z}^{11\\cdot 3}}=\\frac{64}{{z}^{33}}[\/latex]<\/li>\n<li>[latex]\\Large{\\left(\\frac{p}{{q}^{3}}\\right)}^{6}=\\frac{{\\left(p\\right)}^{6}}{{\\left({q}^{3}\\right)}^{6}}=\\frac{{p}^{1\\cdot 6}}{{q}^{3\\cdot 6}}=\\frac{{p}^{6}}{{q}^{18}}[\/latex]<\/li>\n<li>[latex]\\Large{\\left(\\frac{-1}{{t}^{2}}\\right)}^{27}=\\frac{{\\left(-1\\right)}^{27}}{{\\left({t}^{2}\\right)}^{27}}=\\frac{-1}{{t}^{2\\cdot 27}}=\\frac{-1}{{t}^{54}}=-\\frac{1}{{t}^{54}}[\/latex]<\/li>\n<li>[latex]\\Large{\\left({j}^{3}{k}^{-2}\\right)}^{4}={\\left(\\frac{{j}^{3}}{{k}^{2}}\\right)}^{4}=\\frac{{\\left({j}^{3}\\right)}^{4}}{{\\left({k}^{2}\\right)}^{4}}=\\frac{{j}^{3\\cdot 4}}{{k}^{2\\cdot 4}}=\\frac{{j}^{12}}{{k}^{8}}[\/latex]<\/li>\n<li>[latex]\\Large{\\left({m}^{-2}{n}^{-2}\\right)}^{3}={\\left(\\frac{1}{{m}^{2}{n}^{2}}\\right)}^{3}=\\frac{{\\left(1\\right)}^{3}}{{\\left({m}^{2}{n}^{2}\\right)}^{3}}=\\frac{1}{{\\left({m}^{2}\\right)}^{3}{\\left({n}^{2}\\right)}^{3}}=\\frac{1}{{m}^{2\\cdot 3}\\cdot {n}^{2\\cdot 3}}=\\frac{1}{{m}^{6}{n}^{6}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #000000;\">The following video provides more examples of simplifying expressions using the power of a quotient and other exponent rules.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Expressions Using Exponent Rules (Power of a Quotient)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BoBe31pRxFM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<ul>\n<li>Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.<\/li>\n<li>The product rule for exponents:\u00a0For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/li>\n<li>The quotient rule for exponents:\u00a0For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/li>\n<li>The power rule for exponents:\n<ol>\n<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <em>n<\/em>, [latex]\\left(ab\\right)^{n}=a^{n}\\cdot{b^{n}}[\/latex].<\/li>\n<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <em>n<\/em>, [latex]\\displaystyle {\\left(\\frac{a}{b}\\right)}^{n}=\\frac{a^{n}}{b^{n}}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ul>\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-2212\">\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>Simplify Expressions Using Exponent Rules (Power of a Product). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/p-2UkpJQWpo\">https:\/\/youtu.be\/p-2UkpJQWpo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Expressions Using Exponent Rules (Power of a Quotient). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BoBe31pRxFM\">https:\/\/youtu.be\/BoBe31pRxFM<\/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, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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@3.278:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify Expressions Using Exponent Rules (Power of a Product)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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