{"id":2187,"date":"2022-05-18T16:49:47","date_gmt":"2022-05-18T16:49:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2187"},"modified":"2026-01-22T19:54:54","modified_gmt":"2026-01-22T19:54:54","slug":"5-3-properties-of-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/5-3-properties-of-exponents\/","title":{"raw":"5.3.1: Properties of Exponents: Product, Quotient and Power Rules","rendered":"5.3.1: Properties of Exponents: Product, Quotient and Power Rules"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-253\" class=\"standard post-253 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Apply the product rule for exponents.<\/li>\r\n \t<li>Apply the quotient rule for exponents.<\/li>\r\n \t<li>Apply the power rule for exponents.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Product Rule for Exponents<\/h2>\r\nConsider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, [latex]x[\/latex], but they are raised to different exponents. Let's expand each expression, and then rewrite the resulting expression using exponents:\r\n<div style=\"text-align: center;\">[latex]\\begin{align}x^{3}\\cdot x^{4}&amp;=\\stackrel{\\text{3 factors }}{(x\\cdot x\\cdot x)} \\stackrel{\\text{ 4 factors}}{(x\\cdot x\\cdot x\\cdot x)} \\\\ &amp; =\\stackrel{7 \\text{ factors}}{x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x} \\\\ &amp; =x^{7}\\end{align}[\/latex]<\/div>\r\nThe result is that [latex]{x}^{3}\\cdot {x}^{4}={x}^{3+4}={x}^{7}[\/latex].\r\n\r\nNotice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the\u00a0<strong><em>product rule for exponents.<\/em><\/strong>\r\n<div style=\"text-align: center;\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>The Product Rule for Exponents<\/h3>\r\nFor any real numbers [latex]a, m[\/latex] and [latex]n[\/latex], the product rule of exponents states that\u00a0 [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]\r\n<div><\/div>\r\n<div><\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nSimplify the expressions.\r\n<ol>\r\n \t<li>[latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(-3\\right)}^{5}\\cdot \\left(-3\\right)={\\left(-3\\right)}^{5}\\cdot {\\left(-3\\right)}^{1}={\\left(-3\\right)}^{5+1}={\\left(-3\\right)}^{6}[\/latex]<\/li>\r\n \t<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/li>\r\n<\/ol>\r\n<div><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nSimplify each expression.\r\n<ol>\r\n \t<li>[latex]{k}^{6}\\cdot {k}^{9}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{4}\\cdot \\left(\\dfrac{2}{y}\\right)[\/latex]<\/li>\r\n \t<li>[latex]{t}^{3}\\cdot {t}^{6}\\cdot {t}^{5}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"342196\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"342196\"]\r\n<ol>\r\n \t<li>[latex]{k}^{15}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{5}[\/latex]<\/li>\r\n \t<li>[latex]{t}^{14}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 2<\/h3>\r\n<iframe id=\"ohm1961\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1961&amp;theme=oea&amp;iframe_resize_id=ohm1961&amp;show_question_numbers\" width=\"100%\" height=\"150\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\nThe following video shows more examples of how to use the product rule to simplify an expression with exponents.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/P0UVIMy2nuI?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>The Quotient Rule for Exponents<\/h2>\r\nThe\u00a0<em><strong>quotient rule for exponents<\/strong>\u00a0<\/em>allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as [latex]\\dfrac{{y}^{m}}{{y}^{n}}[\/latex]. Consider the example [latex]\\dfrac{{y}^{9}}{{y}^{5}}[\/latex]. Perform the division by canceling common factors.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{y^{9}}{y^{5}} &amp;=\\frac{y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y}{y\\cdot y\\cdot y\\cdot y\\cdot y} \\\\[1mm] &amp;=\\frac{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot y\\cdot y\\cdot y\\cdot y}{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}} \\\\[1mm] &amp; =\\frac{y\\cdot y\\cdot y\\cdot y}{1} \\\\[1mm] &amp; =y^{4}\\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n<div><\/div>\r\n<div>\r\n<div class=\"textbox\">Remember that when we cancel [latex]\\dfrac{y}{y}[\/latex], we are simplifying by division and [latex]\\dfrac {y}{y}=1[\/latex] as long as [latex]y\\neq0[\/latex].<\/div>\r\n<\/div>\r\nNotice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.\r\n<div style=\"text-align: center;\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\r\nIn other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>The Quotient Rule for Exponents<\/h3>\r\nFor any real numbers [latex]a, m[\/latex] and [latex]n[\/latex], provided [latex]a\\neq0[\/latex], the quotient rule of exponents states that\u00a0 [latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]\r\n<div><\/div>\r\n<div><\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nSimplify the expressions.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nUse the quotient rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nWrite each of the following quotients with a single base. Do not simplify further.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{s}^{75}}{{s}^{68}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(-3\\right)}^{6}}{-3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(e{f}^{2}\\right)}^{5}}{{\\left(e{f}^{2}\\right)}^{3}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"544400\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"544400\"]\r\n<ol>\r\n \t<li>[latex]{s}^{7}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(-3\\right)}^{5}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(e{f}^{2}\\right)}^{2}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 4<\/h3>\r\n<iframe id=\"ohm109745\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&amp;theme=oea&amp;iframe_resize_id=ohm109745&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 5<\/h3>\r\n<iframe id=\"ohm109748\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109748&amp;theme=oea&amp;iframe_resize_id=ohm109748&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\nWatch this video to see more examples of how to use the quotient rule for exponents.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/xy6WW7y_GcU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>The Power Rule for Exponents<\/h2>\r\nSuppose an exponential expression is raised to some power.\u00a0 To simplify such an expression, we use the\u00a0<strong><em>power rule of exponents<\/em><\/strong>. Consider the expression [latex]{\\left({x}^{2}\\right)}^{3}[\/latex]. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} {\\left({x}^{2}\\right)}^{3}&amp; = \\stackrel{{3\\text{ factors}}}{{{\\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)}}} \\\\ &amp; = \\stackrel{{3\\text{ factors}}}{\\overbrace{{\\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)}}}\\\\ &amp; = x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\hfill \\\\ &amp; = {x}^{6} \\end{align}[\/latex]<\/div>\r\nThe exponent of the answer is the product of the exponents: [latex]{\\left({x}^{2}\\right)}^{3}={x}^{2\\cdot 3}={x}^{6}[\/latex] because there are three groups of two [latex]x[\/latex]s multiplied together. Therefore, the total number of [latex]x[\/latex]s multiplied together is [latex]3 \\times 2 = 6[\/latex]. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.\r\n<div style=\"text-align: center;\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>The Power Rule of Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and positive integers [latex]m[\/latex] and [latex]n[\/latex], the power rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<\/div>\r\nBe careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, we multiply the exponents.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"5\">Product Rule<\/th>\r\n<th style=\"text-align: center;\" colspan=\"6\">Power Rule<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]5^{3}\\cdot5^{4}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>\u00a0[latex]5^{3+4}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]5^{7}[\/latex]<\/td>\r\n<td>but<\/td>\r\n<td>[latex]\\left(5^{3}\\right)^{4}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]5^{3\\cdot4}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]5^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x^{5}\\cdot x^{2}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]x^{5+2}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]x^{7}[\/latex]<\/td>\r\n<td>but<\/td>\r\n<td>[latex]\\left(x^{5}\\right)^{2}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>\u00a0[latex]x^{5\\cdot2}[\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]x^{10}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(3a\\right)^{7}\\cdot\\left(3a\\right)^{10} [\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]\\left(3a\\right)^{7+10} [\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]\\left(3a\\right)^{17}[\/latex]<\/td>\r\n<td>but<\/td>\r\n<td>[latex]\\left(\\left(3a\\right)^{7}\\right)^{10} [\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]\\left(3a\\right)^{7\\cdot10} [\/latex]<\/td>\r\n<td>=<\/td>\r\n<td>[latex]\\left(3a\\right)^{70}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nSimplify the expressions.\r\n<ol>\r\n \t<li>[latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nWe can apply the power rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]{\\left({x}^{2}\\right)}^{7}={x}^{2\\cdot 7}={x}^{14}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}={\\left(2t\\right)}^{5\\cdot 3}={\\left(2t\\right)}^{15}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}={\\left(-3\\right)}^{5\\cdot 11}={\\left(-3\\right)}^{55}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 6<\/h3>\r\nSimplify the expressions.\r\n<ol>\r\n \t<li>[latex]{\\left({\\left(3y\\right)}^{8}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({t}^{5}\\right)}^{7}[\/latex]<\/li>\r\n \t<li>[latex]{\\left({\\left(-g\\right)}^{4}\\right)}^{4}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"623184\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"623184\"]\r\n<ol>\r\n \t<li>[latex]{\\left(3y\\right)}^{24}[\/latex]<\/li>\r\n \t<li>[latex]{t}^{35}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(-g\\right)}^{16}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 7<\/h3>\r\n<iframe id=\"ohm93370\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93370&amp;theme=oea&amp;iframe_resize_id=ohm93370&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>TRY IT 8<\/h3>\r\n<iframe id=\"ohm93402\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93402&amp;theme=oea&amp;iframe_resize_id=ohm93402&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\nThe following video shows more examples of using the power rule to simplify expressions with exponents.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/VjcKU5rA7F8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-253\" class=\"standard post-253 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Apply the product rule for exponents.<\/li>\n<li>Apply the quotient rule for exponents.<\/li>\n<li>Apply the power rule for exponents.<\/li>\n<\/ul>\n<\/div>\n<h2>The Product Rule for Exponents<\/h2>\n<p>Consider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, [latex]x[\/latex], but they are raised to different exponents. Let&#8217;s expand each expression, and then rewrite the resulting expression using exponents:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}x^{3}\\cdot x^{4}&=\\stackrel{\\text{3 factors }}{(x\\cdot x\\cdot x)} \\stackrel{\\text{ 4 factors}}{(x\\cdot x\\cdot x\\cdot x)} \\\\ & =\\stackrel{7 \\text{ factors}}{x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x} \\\\ & =x^{7}\\end{align}[\/latex]<\/div>\n<p>The result is that [latex]{x}^{3}\\cdot {x}^{4}={x}^{3+4}={x}^{7}[\/latex].<\/p>\n<p>Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the\u00a0<strong><em>product rule for exponents.<\/em><\/strong><\/p>\n<div style=\"text-align: center;\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<div><\/div>\n<div class=\"textbox shaded\">\n<h3>The Product Rule for Exponents<\/h3>\n<p>For any real numbers [latex]a, m[\/latex] and [latex]n[\/latex], the product rule of exponents states that\u00a0 [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/p>\n<div><\/div>\n<div><\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Simplify the expressions.<\/p>\n<ol>\n<li>[latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\n<li>[latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\n<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\n<li>[latex]{\\left(-3\\right)}^{5}\\cdot \\left(-3\\right)={\\left(-3\\right)}^{5}\\cdot {\\left(-3\\right)}^{1}={\\left(-3\\right)}^{5+1}={\\left(-3\\right)}^{6}[\/latex]<\/li>\n<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/li>\n<\/ol>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Simplify each expression.<\/p>\n<ol>\n<li>[latex]{k}^{6}\\cdot {k}^{9}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{4}\\cdot \\left(\\dfrac{2}{y}\\right)[\/latex]<\/li>\n<li>[latex]{t}^{3}\\cdot {t}^{6}\\cdot {t}^{5}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q342196\">Show Answer<\/span><\/p>\n<div id=\"q342196\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{k}^{15}[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{2}{y}\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{t}^{14}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 2<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1961\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1961&amp;theme=oea&amp;iframe_resize_id=ohm1961&amp;show_question_numbers\" width=\"100%\" height=\"150\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<p>The following video shows more examples of how to use the product rule to simplify an expression with exponents.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/P0UVIMy2nuI?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Quotient Rule for Exponents<\/h2>\n<p>The\u00a0<em><strong>quotient rule for exponents<\/strong>\u00a0<\/em>allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as [latex]\\dfrac{{y}^{m}}{{y}^{n}}[\/latex]. Consider the example [latex]\\dfrac{{y}^{9}}{{y}^{5}}[\/latex]. Perform the division by canceling common factors.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{y^{9}}{y^{5}} &=\\frac{y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y\\cdot y}{y\\cdot y\\cdot y\\cdot y\\cdot y} \\\\[1mm] &=\\frac{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot y\\cdot y\\cdot y\\cdot y}{\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}\\cdot\\cancel{y}} \\\\[1mm] & =\\frac{y\\cdot y\\cdot y\\cdot y}{1} \\\\[1mm] & =y^{4}\\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<div><\/div>\n<div>\n<div class=\"textbox\">Remember that when we cancel [latex]\\dfrac{y}{y}[\/latex], we are simplifying by division and [latex]\\dfrac {y}{y}=1[\/latex] as long as [latex]y\\neq0[\/latex].<\/div>\n<\/div>\n<p>Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<p>In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Quotient Rule for Exponents<\/h3>\n<p>For any real numbers [latex]a, m[\/latex] and [latex]n[\/latex], provided [latex]a\\neq0[\/latex], the quotient rule of exponents states that\u00a0 [latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/p>\n<div><\/div>\n<div><\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Simplify the expressions.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>Use the quotient rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Write each of the following quotients with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]\\dfrac{{s}^{75}}{{s}^{68}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(-3\\right)}^{6}}{-3}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(e{f}^{2}\\right)}^{5}}{{\\left(e{f}^{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=\"q544400\">Show Answer<\/span><\/p>\n<div id=\"q544400\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{s}^{7}[\/latex]<\/li>\n<li>[latex]{\\left(-3\\right)}^{5}[\/latex]<\/li>\n<li>[latex]{\\left(e{f}^{2}\\right)}^{2}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 4<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109745\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&amp;theme=oea&amp;iframe_resize_id=ohm109745&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 5<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm109748\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109748&amp;theme=oea&amp;iframe_resize_id=ohm109748&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of how to use the quotient rule for exponents.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/xy6WW7y_GcU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Power Rule for Exponents<\/h2>\n<p>Suppose an exponential expression is raised to some power.\u00a0 To simplify such an expression, we use the\u00a0<strong><em>power rule of exponents<\/em><\/strong>. Consider the expression [latex]{\\left({x}^{2}\\right)}^{3}[\/latex]. The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} {\\left({x}^{2}\\right)}^{3}& = \\stackrel{{3\\text{ factors}}}{{{\\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)\\cdot \\left({x}^{2}\\right)}}} \\\\ & = \\stackrel{{3\\text{ factors}}}{\\overbrace{{\\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)\\cdot \\left(\\stackrel{{2\\text{ factors}}}{{\\overbrace{x\\cdot x}}}\\right)}}}\\\\ & = x\\cdot x\\cdot x\\cdot x\\cdot x\\cdot x\\hfill \\\\ & = {x}^{6} \\end{align}[\/latex]<\/div>\n<p>The exponent of the answer is the product of the exponents: [latex]{\\left({x}^{2}\\right)}^{3}={x}^{2\\cdot 3}={x}^{6}[\/latex] because there are three groups of two [latex]x[\/latex]s multiplied together. Therefore, the total number of [latex]x[\/latex]s multiplied together is [latex]3 \\times 2 = 6[\/latex]. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents.<\/p>\n<div style=\"text-align: center;\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>The Power Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and positive integers [latex]m[\/latex] and [latex]n[\/latex], the power rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/div>\n<div><\/div>\n<div><\/div>\n<\/div>\n<p>Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, we multiply the exponents.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"5\">Product Rule<\/th>\n<th style=\"text-align: center;\" colspan=\"6\">Power Rule<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]5^{3}\\cdot5^{4}[\/latex]<\/td>\n<td>=<\/td>\n<td>\u00a0[latex]5^{3+4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{7}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(5^{3}\\right)^{4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{3\\cdot4}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]5^{12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x^{5}\\cdot x^{2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{5+2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{7}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(x^{5}\\right)^{2}[\/latex]<\/td>\n<td>=<\/td>\n<td>\u00a0[latex]x^{5\\cdot2}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]x^{10}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(3a\\right)^{7}\\cdot\\left(3a\\right)^{10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{7+10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{17}[\/latex]<\/td>\n<td>but<\/td>\n<td>[latex]\\left(\\left(3a\\right)^{7}\\right)^{10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{7\\cdot10}[\/latex]<\/td>\n<td>=<\/td>\n<td>[latex]\\left(3a\\right)^{70}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Simplify the expressions.<\/p>\n<ol>\n<li>[latex]{\\left({x}^{2}\\right)}^{7}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>We can apply the power rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left({x}^{2}\\right)}^{7}={x}^{2\\cdot 7}={x}^{14}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(2t\\right)}^{5}\\right)}^{3}={\\left(2t\\right)}^{5\\cdot 3}={\\left(2t\\right)}^{15}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(-3\\right)}^{5}\\right)}^{11}={\\left(-3\\right)}^{5\\cdot 11}={\\left(-3\\right)}^{55}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 6<\/h3>\n<p>Simplify the expressions.<\/p>\n<ol>\n<li>[latex]{\\left({\\left(3y\\right)}^{8}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left({t}^{5}\\right)}^{7}[\/latex]<\/li>\n<li>[latex]{\\left({\\left(-g\\right)}^{4}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q623184\">Show Answer<\/span><\/p>\n<div id=\"q623184\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{\\left(3y\\right)}^{24}[\/latex]<\/li>\n<li>[latex]{t}^{35}[\/latex]<\/li>\n<li>[latex]{\\left(-g\\right)}^{16}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 7<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm93370\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93370&amp;theme=oea&amp;iframe_resize_id=ohm93370&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>TRY IT 8<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm93402\" class=\"resizable\" title=\"Interactive problem\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93402&amp;theme=oea&amp;iframe_resize_id=ohm93402&amp;show_question_numbers\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<p>The following video shows more examples of using the power rule to simplify expressions with exponents.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/VjcKU5rA7F8?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-2187\">\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>Adaptation and Revision. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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><\/li><li>Product Rule for Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P0UVIMy2nuI\">https:\/\/youtu.be\/P0UVIMy2nuI<\/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>Quotient Rule for Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xy6WW7y_GcU\">https:\/\/youtu.be\/xy6WW7y_GcU<\/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>Using the Power Rule to Simplify Expressions With Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/VjcKU5rA7F8\">https:\/\/youtu.be\/VjcKU5rA7F8<\/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>Question ID 109745, 109748. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 93370, 93399, 93402. <strong>Authored by<\/strong>: Michael Jenck. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1961. <strong>Authored by<\/strong>: David Lippman. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax 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><\/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":422608,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Adaptation and Revision\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley 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