{"id":44,"date":"2023-06-21T13:22:27","date_gmt":"2023-06-21T13:22:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/rules-for-exponents\/"},"modified":"2023-07-04T04:53:49","modified_gmt":"2023-07-04T04:53:49","slug":"rules-for-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/rules-for-exponents\/","title":{"raw":"\u25aa   Rules for Exponents","rendered":"\u25aa   Rules for Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the product rule for exponents.<\/li>\r\n \t<li>Use the quotient rule for exponents.<\/li>\r\n \t<li>Use the power rule for exponents.<\/li>\r\n<\/ul>\r\n<\/div>\r\nConsider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, <em>x<\/em>, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.\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 <em>product rule of exponents.<\/em>\r\n<div style=\"text-align: center;\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\r\nNow consider an example with real numbers.\r\n<div style=\"text-align: center;\">[latex]{2}^{3}\\cdot {2}^{4}={2}^{3+4}={2}^{7}[\/latex]<\/div>\r\nWe can always check that this is true by simplifying each exponential expression. We find that [latex]{2}^{3}[\/latex] is 8, [latex]{2}^{4}[\/latex] is 16, and [latex]{2}^{7}[\/latex] is 128. The product [latex]8\\cdot 16[\/latex] equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Product Rule of Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], the product rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Product Rule<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\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[reveal-answer q=\"878162\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"878162\"]\r\nUse the product rule to simplify each expression.\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}[\/latex]<\/li>\r\n<\/ol>\r\nAt first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.\r\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}=\\left({x}^{2}\\cdot {x}^{5}\\right)\\cdot {x}^{3}=\\left({x}^{2+5}\\right)\\cdot {x}^{3}={x}^{7}\\cdot {x}^{3}={x}^{7+3}={x}^{10}[\/latex]<\/div>\r\nNotice we get the same result by adding the three exponents in one step.\r\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\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=\"562258\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"562258\"]\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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1961&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to use the product rule to simplify an expression with exponents.\r\n\r\nhttps:\/\/youtu.be\/P0UVIMy2nuI\r\n<h2>Using the Quotient Rule of Exponents<\/h2>\r\nThe <em>quotient rule of exponents<\/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], where [latex]m&gt;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>\r\n<div class=\"textbox examples\">\r\n<h3>Recall<\/h3>\r\nWhen simplifying fractions, \"canceling out\" always leaves a 1 behind because [latex]\\dfrac{a}{a}=1[\/latex].\r\n\r\n<\/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 style=\"text-align: center;\">[latex]\\dfrac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[\/latex]<\/div>\r\nFor the time being, we must be aware of the condition [latex]m&gt;n[\/latex]. Otherwise, the difference [latex]m-n[\/latex] could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Quotient Rule of Exponents<\/h3>\r\nFor any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], such that [latex]m&gt;n[\/latex], the quotient rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Quotient Rule<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\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[reveal-answer q=\"717838\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"717838\"]\r\n\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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite each of the following products 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=\"677916\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"677916\"]\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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&amp;theme=oea&amp;iframe_resize_id=mom60[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109748&amp;theme=oea&amp;iframe_resize_id=mom70[\/embed]\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\nhttps:\/\/youtu.be\/xy6WW7y_GcU\r\n<h2>Using the Power Rule of Exponents<\/h2>\r\nSuppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the <em>power rule of exponents<\/em>. 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]. 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\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, you 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+1-} [\/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\">\r\n<h3>A General Note: 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>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Power Rule<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\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[reveal-answer q=\"992335\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"992335\"]\r\n\r\nUse 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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\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=\"875151\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875151\"]\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[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93370&amp;theme=oea&amp;iframe_resize_id=mom80[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93399&amp;theme=oea&amp;iframe_resize_id=mom90[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93402&amp;theme=oea&amp;iframe_resize_id=mom100[\/embed]\r\n\r\n<\/div>\r\nThe following video gives more examples of using the power rule to simplify expressions with exponents.\r\nhttps:\/\/youtu.be\/VjcKU5rA7F8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the product rule for exponents.<\/li>\n<li>Use the quotient rule for exponents.<\/li>\n<li>Use the power rule for exponents.<\/li>\n<\/ul>\n<\/div>\n<p>Consider the product [latex]{x}^{3}\\cdot {x}^{4}[\/latex]. Both terms have the same base, <em>x<\/em>, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.<\/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 <em>product rule of exponents.<\/em><\/p>\n<div style=\"text-align: center;\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<p>Now consider an example with real numbers.<\/p>\n<div style=\"text-align: center;\">[latex]{2}^{3}\\cdot {2}^{4}={2}^{3+4}={2}^{7}[\/latex]<\/div>\n<p>We can always check that this is true by simplifying each exponential expression. We find that [latex]{2}^{3}[\/latex] is 8, [latex]{2}^{4}[\/latex] is 16, and [latex]{2}^{7}[\/latex] is 128. The product [latex]8\\cdot 16[\/latex] equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Product Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], the product rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q878162\">Show Solution<\/span><\/p>\n<div id=\"q878162\" class=\"hidden-answer\" style=\"display: none\">\nUse the product rule to simplify each expression.<\/p>\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}[\/latex]<\/li>\n<\/ol>\n<p>At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.<\/p>\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}=\\left({x}^{2}\\cdot {x}^{5}\\right)\\cdot {x}^{3}=\\left({x}^{2+5}\\right)\\cdot {x}^{3}={x}^{7}\\cdot {x}^{3}={x}^{7+3}={x}^{10}[\/latex]<\/div>\n<p>Notice we get the same result by adding the three exponents in one step.<\/p>\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/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=\"q562258\">Show Solution<\/span><\/p>\n<div id=\"q562258\" 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<p><iframe loading=\"lazy\" id=\"ohm1961\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1961&#38;theme=oea&#38;iframe_resize_id=ohm1961&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to use the product rule to simplify an expression with exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Expressions Using the Product Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P0UVIMy2nuI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Quotient Rule of Exponents<\/h2>\n<p>The <em>quotient rule of exponents<\/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], where [latex]m>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>\n<div class=\"textbox examples\">\n<h3>Recall<\/h3>\n<p>When simplifying fractions, &#8220;canceling out&#8221; always leaves a 1 behind because [latex]\\dfrac{a}{a}=1[\/latex].<\/p>\n<\/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 style=\"text-align: center;\">[latex]\\dfrac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[\/latex]<\/div>\n<p>For the time being, we must be aware of the condition [latex]m>n[\/latex]. Otherwise, the difference [latex]m-n[\/latex] could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule of Exponents<\/h3>\n<p>For any real number [latex]a[\/latex] and natural numbers [latex]m[\/latex] and [latex]n[\/latex], such that [latex]m>n[\/latex], the quotient rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q717838\">Show Solution<\/span><\/p>\n<div id=\"q717838\" class=\"hidden-answer\" style=\"display: none\">\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 key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products 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=\"q677916\">Show Solution<\/span><\/p>\n<div id=\"q677916\" 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<p><iframe loading=\"lazy\" id=\"ohm109745\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109745&#38;theme=oea&#38;iframe_resize_id=ohm109745&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm109748\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109748&#38;theme=oea&#38;iframe_resize_id=ohm109748&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/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\" id=\"oembed-2\" title=\"Simplify Expressions Using the Quotient Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xy6WW7y_GcU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using the Power Rule of Exponents<\/h2>\n<p>Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the <em>power rule of exponents<\/em>. 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]. 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<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, you 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+1-}[\/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\">\n<h3>A General Note: 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>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Power Rule<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992335\">Show Solution<\/span><\/p>\n<div id=\"q992335\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use 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>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/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=\"q875151\">Show Solution<\/span><\/p>\n<div id=\"q875151\" 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<p><iframe loading=\"lazy\" id=\"ohm93370\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93370&#38;theme=oea&#38;iframe_resize_id=ohm93370&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm93399\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93399&#38;theme=oea&#38;iframe_resize_id=ohm93399&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm93402\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93402&#38;theme=oea&#38;iframe_resize_id=ohm93402&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video gives more examples of using the power rule to simplify expressions with exponents.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Expressions Using the Power Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VjcKU5rA7F8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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-44\">\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><\/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>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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(Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/VjcKU5rA7F8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 109745, 109748\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 93370, 93399, 93402\",\"author\":\"Michael Jenck\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 1961\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"1cff3a71-5ed2-4bf4-badf-d6eceaf22a0a","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-44","chapter","type-chapter","status-publish","hentry"],"part":31,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/44","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions"}],"predecessor-version":[{"id":835,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions\/835"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/31"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/44\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=44"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=44"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=44"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}