{"id":18559,"date":"2022-04-21T22:32:45","date_gmt":"2022-04-21T22:32:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18559"},"modified":"2022-04-28T23:17:48","modified_gmt":"2022-04-28T23:17:48","slug":"cr-21-laws-of-exponents-positive-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-21-laws-of-exponents-positive-exponents\/","title":{"raw":"CR.21: Laws of Exponents (Positive Exponents)","rendered":"CR.21: Laws of Exponents (Positive 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 \t<li>Simplify expressions with exponents equal to zero.<\/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\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Power Rule<\/h3>\r\nSimplify each expression.\r\n<ol>\r\n \t<li>[latex]{\\left(-2{x}\\right)}^{9}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(2{x}^{3}{y}\\right)}^{7}[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\dfrac{6{x}^{3}}{7{y}^{5}}\\right)^2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"992340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"992340\"]\r\n\r\nUse the power rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]{\\left(-1\\cdot2\\cdot{x}\\right)}^{9}={\\left(-1\\right)}^{9}\\cdot{\\left(2\\right)}^{9}\\cdot{\\left(x\\right)}^{9}=-1\\cdot512\\cdot{x}^{9}=-512{x}^{9}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(2\\right)}^{7}\\cdot{\\left({x}^{3}\\right)}^{7}\\cdot{y}^{7}=128\\cdot{x}^{3\\cdot7}\\cdot{y}^{7}=128{x}^{21}{y}^{7}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{\\left(6{x}^{3}\\right)^2}{\\left(7{y}^{5}\\right)^2}=\\dfrac{6^2\\cdot{x}^{3\\cdot2}}{7^2\\cdot{y}^{5\\cdot2}}=\\dfrac{36x^6}{49y^{10}}[\/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\nSimplify each expression.\r\n<ol>\r\n \t<li>[latex]{\\left(-5{x}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(3{x}^{4}{y}\\right)}^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(\\dfrac{3{x}^{5}}{5{y}^{4}}\\right)^2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"885150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"885150\"]\r\n<ol>\r\n \t<li>[latex]-125{x}^{3}[\/latex]<\/li>\r\n \t<li>[latex]9{x}^{8}{y}^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{9x^{10}}{25y^8}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\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\r\n<h2>Zero Exponents<\/h2>\r\nReturn to the quotient rule. We made the condition that [latex]m&gt;n[\/latex] so that the difference [latex]m-n[\/latex] would never be zero or negative. What would happen if [latex]m=n[\/latex]? In this case, we would use the <em>zero exponent rule of exponents<\/em> to simplify the expression to 1. To see how this is done, let us begin with an example.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{t^{8}}{t^{8}}=\\dfrac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\r\nIf we were to simplify the original expression using the quotient rule, we would have\r\n<div style=\"text-align: center;\">[latex]\\dfrac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/div>\r\nIf we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.\r\n<div style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/div>\r\nThe sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Zero Exponent Rule of Exponents<\/h3>\r\nFor any nonzero real number [latex]a[\/latex], the zero exponent rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>using order of operations with fractions<\/h3>\r\nWhen simplifying expressions with exponents, it is sometimes helpful to rely on the rule for multiplying fractions to separate the factors before doing work on them. For example, to simplify the expression\u00a0[latex]\\dfrac{5 a^m z^2}{a^mz}[\/latex]\u00a0using exponent rules, you may find it helpful to break the fraction up into a product of fractions, then simplify.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{5 a^m z^2}{a^mz}\\quad=\\quad 5\\cdot\\dfrac{a^m}{a^m}\\cdot\\dfrac{z^2}{z} \\quad=\\quad 5 \\cdot a^{m-m}\\cdot z^{2-1}\\quad=\\quad 5\\cdot a^0 \\cdot z^1 \\quad=\\quad 5z[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Zero Exponent Rule<\/h3>\r\nSimplify each expression using the zero exponent rule of exponents.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{c}^{3}}{{c}^{3}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{-3{x}^{5}}{{x}^{5}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"913171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"913171\"]\r\nUse the zero exponent and other rules to simplify each expression.\r\n<ol>\r\n \t<li>[latex]\\begin{align}\\frac{c^{3}}{c^{3}} &amp; =c^{3-3} \\\\ &amp; =c^{0} \\\\ &amp; =1\\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\frac{-3{x}^{5}}{{x}^{5}}&amp; = -3\\cdot \\frac{{x}^{5}}{{x}^{5}} \\\\ &amp; = -3\\cdot {x}^{5 - 5} \\\\ &amp; = -3\\cdot {x}^{0} \\\\ &amp; = -3\\cdot 1 \\\\ &amp; = -3 \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}&amp; = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{1+3}} &amp;&amp; \\text{Use the product rule in the denominator}. \\\\ &amp; = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{4}} &amp;&amp; \\text{Simplify}. \\\\ &amp; = {\\left({j}^{2}k\\right)}^{4 - 4} &amp;&amp; \\text{Use the quotient rule}. \\\\ &amp; = {\\left({j}^{2}k\\right)}^{0} &amp;&amp; \\text{Simplify}. \\\\ &amp; = 1 \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align} \\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}&amp; = 5{\\left(r{s}^{2}\\right)}^{2 - 2} &amp;&amp; \\text{Use the quotient rule}. \\\\ &amp; = 5{\\left(r{s}^{2}\\right)}^{0} &amp;&amp; \\text{Simplify}. \\\\ &amp; = 5\\cdot 1 &amp;&amp; \\text{Use the zero exponent rule}. \\\\ &amp; = 5 &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify each expression using the zero exponent rule of exponents.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{t}^{7}}{{t}^{7}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(d{e}^{2}\\right)}^{11}}{2{\\left(d{e}^{2}\\right)}^{11}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{w}^{4}\\cdot {w}^{2}}{{w}^{6}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{3}\\cdot {t}^{4}}{{t}^{2}\\cdot {t}^{5}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"703483\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"703483\"]\r\n<ol>\r\n \t<li>[latex]1[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]1[\/latex]<\/li>\r\n \t<li>[latex]1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=44120&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7833&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\nIn this video we show more examples of how to simplify expressions with zero exponents.\r\n\r\nhttps:\/\/youtu.be\/rpoUg32utlc","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<li>Simplify expressions with exponents equal to zero.<\/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<div class=\"textbox exercises\">\n<h3>Example: Using the Power Rule<\/h3>\n<p>Simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left(-2{x}\\right)}^{9}[\/latex]<\/li>\n<li>[latex]{\\left(2{x}^{3}{y}\\right)}^{7}[\/latex]<\/li>\n<li>[latex]\\left(\\dfrac{6{x}^{3}}{7{y}^{5}}\\right)^2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q992340\">Show Solution<\/span><\/p>\n<div id=\"q992340\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the power rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left(-1\\cdot2\\cdot{x}\\right)}^{9}={\\left(-1\\right)}^{9}\\cdot{\\left(2\\right)}^{9}\\cdot{\\left(x\\right)}^{9}=-1\\cdot512\\cdot{x}^{9}=-512{x}^{9}[\/latex]<\/li>\n<li>[latex]{\\left(2\\right)}^{7}\\cdot{\\left({x}^{3}\\right)}^{7}\\cdot{y}^{7}=128\\cdot{x}^{3\\cdot7}\\cdot{y}^{7}=128{x}^{21}{y}^{7}[\/latex]<\/li>\n<li>[latex]\\dfrac{\\left(6{x}^{3}\\right)^2}{\\left(7{y}^{5}\\right)^2}=\\dfrac{6^2\\cdot{x}^{3\\cdot2}}{7^2\\cdot{y}^{5\\cdot2}}=\\dfrac{36x^6}{49y^{10}}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify each expression.<\/p>\n<ol>\n<li>[latex]{\\left(-5{x}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]{\\left(3{x}^{4}{y}\\right)}^{2}[\/latex]<\/li>\n<li>[latex]\\left(\\dfrac{3{x}^{5}}{5{y}^{4}}\\right)^2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q885150\">Show Solution<\/span><\/p>\n<div id=\"q885150\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-125{x}^{3}[\/latex]<\/li>\n<li>[latex]9{x}^{8}{y}^{2}[\/latex]<\/li>\n<li>[latex]\\dfrac{9x^{10}}{25y^8}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\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<h2>Zero Exponents<\/h2>\n<p>Return to the quotient rule. We made the condition that [latex]m>n[\/latex] so that the difference [latex]m-n[\/latex] would never be zero or negative. What would happen if [latex]m=n[\/latex]? In this case, we would use the <em>zero exponent rule of exponents<\/em> to simplify the expression to 1. To see how this is done, let us begin with an example.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{t^{8}}{t^{8}}=\\dfrac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\n<p>If we were to simplify the original expression using the quotient rule, we would have<\/p>\n<div style=\"text-align: center;\">[latex]\\dfrac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/div>\n<p>If we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/div>\n<p>The sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Zero Exponent Rule of Exponents<\/h3>\n<p>For any nonzero real number [latex]a[\/latex], the zero exponent rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>using order of operations with fractions<\/h3>\n<p>When simplifying expressions with exponents, it is sometimes helpful to rely on the rule for multiplying fractions to separate the factors before doing work on them. For example, to simplify the expression\u00a0[latex]\\dfrac{5 a^m z^2}{a^mz}[\/latex]\u00a0using exponent rules, you may find it helpful to break the fraction up into a product of fractions, then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{5 a^m z^2}{a^mz}\\quad=\\quad 5\\cdot\\dfrac{a^m}{a^m}\\cdot\\dfrac{z^2}{z} \\quad=\\quad 5 \\cdot a^{m-m}\\cdot z^{2-1}\\quad=\\quad 5\\cdot a^0 \\cdot z^1 \\quad=\\quad 5z[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Zero Exponent Rule<\/h3>\n<p>Simplify each expression using the zero exponent rule of exponents.<\/p>\n<ol>\n<li>[latex]\\dfrac{{c}^{3}}{{c}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{-3{x}^{5}}{{x}^{5}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}[\/latex]<\/li>\n<li>[latex]\\dfrac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q913171\">Show Solution<\/span><\/p>\n<div id=\"q913171\" class=\"hidden-answer\" style=\"display: none\">\nUse the zero exponent and other rules to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\begin{align}\\frac{c^{3}}{c^{3}} & =c^{3-3} \\\\ & =c^{0} \\\\ & =1\\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{-3{x}^{5}}{{x}^{5}}& = -3\\cdot \\frac{{x}^{5}}{{x}^{5}} \\\\ & = -3\\cdot {x}^{5 - 5} \\\\ & = -3\\cdot {x}^{0} \\\\ & = -3\\cdot 1 \\\\ & = -3 \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{{\\left({j}^{2}k\\right)}^{4}}{\\left({j}^{2}k\\right)\\cdot {\\left({j}^{2}k\\right)}^{3}}& = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{1+3}} && \\text{Use the product rule in the denominator}. \\\\ & = \\frac{{\\left({j}^{2}k\\right)}^{4}}{{\\left({j}^{2}k\\right)}^{4}} && \\text{Simplify}. \\\\ & = {\\left({j}^{2}k\\right)}^{4 - 4} && \\text{Use the quotient rule}. \\\\ & = {\\left({j}^{2}k\\right)}^{0} && \\text{Simplify}. \\\\ & = 1 \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align} \\frac{5{\\left(r{s}^{2}\\right)}^{2}}{{\\left(r{s}^{2}\\right)}^{2}}& = 5{\\left(r{s}^{2}\\right)}^{2 - 2} && \\text{Use the quotient rule}. \\\\ & = 5{\\left(r{s}^{2}\\right)}^{0} && \\text{Simplify}. \\\\ & = 5\\cdot 1 && \\text{Use the zero exponent rule}. \\\\ & = 5 && \\text{Simplify}. \\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify each expression using the zero exponent rule of exponents.<\/p>\n<ol>\n<li>[latex]\\dfrac{{t}^{7}}{{t}^{7}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(d{e}^{2}\\right)}^{11}}{2{\\left(d{e}^{2}\\right)}^{11}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{w}^{4}\\cdot {w}^{2}}{{w}^{6}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{3}\\cdot {t}^{4}}{{t}^{2}\\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=\"q703483\">Show Solution<\/span><\/p>\n<div id=\"q703483\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm44120\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=44120&#38;theme=oea&#38;iframe_resize_id=ohm44120&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm7833\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7833&#38;theme=oea&#38;iframe_resize_id=ohm7833&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In this video we show more examples of how to simplify expressions with zero exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Simplify Expressions Using the Quotient and Zero Exponent Rules\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rpoUg32utlc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"author":264444,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18559","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18559","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18559\/revisions"}],"predecessor-version":[{"id":18716,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18559\/revisions\/18716"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18559\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18559"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18559"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18559"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}