{"id":10850,"date":"2017-06-05T21:31:07","date_gmt":"2017-06-05T21:31:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10850"},"modified":"2021-05-21T21:25:03","modified_gmt":"2021-05-21T21:25:03","slug":"using-the-quotient-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cpcc-algebra-trig-I-support\/chapter\/using-the-quotient-property\/","title":{"raw":"Simplifying Variable Expressions Using Exponent Properties II","rendered":"Simplifying Variable Expressions Using Exponent Properties II"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions using the Quotient Property of Exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplify Expressions Using the Quotient Property of Exponents<\/h2>\r\nEarlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.\r\n<div class=\"textbox shaded\">\r\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\r\nIf [latex]a\\text{ and }b[\/latex] are real numbers and [latex]m\\text{ and }n[\/latex] are whole numbers, then\r\n\r\n[latex]\\begin{array}{cccc}\\text{Product Property}\\hfill &amp; &amp; &amp; \\hfill {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\text{Power Property}\\hfill &amp; &amp; &amp; \\hfill {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\text{Product to a Power}\\hfill &amp; &amp; &amp; \\hfill {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\nNow we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions\u2014which are also quotients.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n\r\n[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]\r\n\r\n<\/div>\r\nAs before, we'll try to discover a property by looking at some examples.\r\n\r\nLet\u2019s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\r\nYou can rewrite the expression as: [latex] \\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of [latex]4[\/latex] in the numerator and denominator: [latex] \\displaystyle [\/latex]\r\n\r\nFinally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, [latex]3[\/latex], is the difference between the two exponents in the original expression, [latex]5[\/latex] and [latex]2[\/latex].\r\n\r\nSo,\u00a0[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].\r\n\r\nNow, let's consider an example in which the base is the variable [latex]x[\/latex].\r\n\r\n[latex]\\begin{array}{cccccccccc}\\text{Consider}\\hfill &amp; &amp; &amp; \\hfill {\\Large\\frac{{x}^{5}}{{x}^{2}}}\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; \\hfill {\\Large\\frac{{x}^{2}}{{x}^{3}}}\\hfill \\\\ \\text{What do they mean?}\\hfill &amp; &amp; &amp; \\hfill {\\Large\\frac{x\\cdot x\\cdot x\\cdot x\\cdot x}{x\\cdot x}}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill {\\Large\\frac{x\\cdot x}{x\\cdot x\\cdot x}}\\hfill \\\\ \\text{Use the Equivalent Fractions Property.}\\hfill &amp; &amp; &amp; \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot x\\cdot x\\cdot x}{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}{\\overline{)x}\\cdot \\overline{)x}\\cdot x}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill {x}^{3}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill {\\Large\\frac{1}{x}}\\hfill \\end{array}[\/latex]\r\n\r\nNotice that in each case the bases were the same and we subtracted the exponents.\u00a0\u00a0So, to divide two exponential terms with the same base, subtract the exponents.\r\n<ul id=\"fs-id916657\">\r\n \t<li>When the larger exponent was in the numerator, we were left with factors in the numerator and [latex]1[\/latex] in the denominator, which we simplified.<\/li>\r\n \t<li>When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[\/latex] in the numerator, which could not be simplified.<\/li>\r\n<\/ul>\r\nWe write:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}\\frac{{x}^{5}}{{x}^{2}}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ {x}^{5 - 2}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{1}{{x}^{3 - 2}}\\hfill \\\\ {x}^{3}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Quotient Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then\r\n\r\n[latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m&gt;n\\text{ and }{\\Large\\frac{{a}^{m}}{{a}^{n}}}={\\Large\\frac{1}{{a}^{n-m}}},n&gt;m[\/latex]\r\n\r\n<\/div>\r\nA couple of examples with numbers may help to verify this property.\r\n\r\n[latex]\\begin{array}{cccc}\\frac{{3}^{4}}{{3}^{2}}\\stackrel{?}{=}{3}^{4 - 2}\\hfill &amp; &amp; &amp; \\hfill \\frac{{5}^{2}}{{5}^{3}}\\stackrel{?}{=}\\frac{1}{{5}^{3 - 2}}\\hfill \\\\ \\frac{81}{9}\\stackrel{?}{=}{3}^{2}\\hfill &amp; &amp; &amp; \\hfill \\frac{25}{125}\\stackrel{?}{=}\\frac{1}{{5}^{1}}\\hfill \\\\ 9=9 \\hfill &amp; &amp; &amp; \\hfill \\frac{1}{5}=\\frac{1}{5}\\hfill \\end{array}[\/latex]\r\n\r\nWhen we work with numbers and the exponent is less than or equal to [latex]3[\/latex], we will apply the exponent. When the exponent is greater than [latex]3[\/latex] , we leave the answer in exponential form.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]\r\n2. [latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168469856047\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 10th over x to the 8th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since 10 &gt; 8, there are more factors of [latex]x[\/latex] in the numerator.<\/td>\r\n<td>[latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]m&gt;n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex] .<\/td>\r\n<td>[latex]{x}^{\\color{red}{10-8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since 9 &gt; 2, there are more factors of 2 in the numerator.<\/td>\r\n<td>[latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]m&gt;n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex].<\/td>\r\n<td>[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that when the larger exponent is in the numerator, we are left with factors in the numerator.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146219[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]\r\n2. [latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]\r\n[reveal-answer q=\"738923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"738923\"]\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168469768330\" class=\"unnumbered unstyled\" summary=\"The first line shows b to the 10th over b to the 15th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">1.<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.34375px;\">\r\n<td style=\"height: 15.34375px;\">Since [latex]15&gt;10[\/latex], there are more factors of [latex]b[\/latex] in the denominator.<\/td>\r\n<td style=\"height: 15.34375px;\">[latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Use the quotient property with [latex]n&gt;m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td style=\"height: 15px;\">[latex]\\Large\\frac{\\color{red}{1}}{{b}^{\\color{red}{15-10}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]\\Large\\frac{1}{{b}^{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469585119\" class=\"unnumbered unstyled\" summary=\"The first line shows 3 to the 3rd over 3 to the 5th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]5&gt;3[\/latex], there are more factors of [latex]3[\/latex] in the denominator.<\/td>\r\n<td>[latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]n&gt;m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{\\color{red}{1}}{{3}^{\\color{red}{5-3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{1}{{3}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Apply the exponent.<\/td>\r\n<td>[latex]\\Large\\frac{1}{9}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146220[\/ohm_question]\r\n\r\n<\/div>\r\nNow let's see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]\r\n2. [latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]\r\n[reveal-answer q=\"903400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903400\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468311860\" class=\"unnumbered unstyled\" summary=\"The first line shows a to the 5th over a to the 9th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15.2344px;\">\r\n<td style=\"height: 15.2344px;\">1.<\/td>\r\n<td style=\"height: 15.2344px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Since [latex]9&gt;5[\/latex], there are more [latex]a[\/latex] 's in the denominator and so we will end up with factors in the denominator.<\/td>\r\n<td style=\"height: 30px;\">[latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Use the Quotient Property for [latex]n&gt;m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td style=\"height: 30px;\">[latex]\\Large\\frac{\\color{red}{1}}{{a}^{\\color{red}{9-5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]\\Large\\frac{1}{{a}^{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468257607\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">2.<\/td>\r\n<td style=\"height: 15px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td style=\"height: 30px;\">Notice there are more factors of [latex]x[\/latex] in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.<\/td>\r\n<td style=\"height: 30px;\">[latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.28125px;\">\r\n<td style=\"height: 15.28125px;\">Use the Quotient Property for [latex]m&gt;n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{n-m}[\/latex].<\/td>\r\n<td style=\"height: 15.28125px;\">[latex]{x}^{\\color{red}{11-7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Simplify.<\/td>\r\n<td style=\"height: 15px;\">[latex]{x}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146889[\/ohm_question]\r\n\r\n<\/div>\r\nWhen dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex][reveal-answer q=\"23604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23604\"]Separate into numerical and variable factors.[latex] \\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]\r\n\r\nSince the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.\r\n\r\n[latex] \\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]\r\n\r\nAnswer\r\n[latex] \\frac{12{{x}^{4}}}{2x}=6{{x}^{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.\r\n\r\nhttps:\/\/youtu.be\/Jmf-CPhm3XM\r\n<h2>Simplify Quotients Raised to a Power<\/h2>\r\nNow we will look at an example that will lead us to the Quotient to a Power Property.\r\n\r\nNow let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex] \\displaystyle \\frac{3}{4}[\/latex] and raise it to the [latex]3<sup>rd<\/sup>[\/latex]power.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\r\nYou can see that raising the quotient to the power of [latex]3[\/latex] can also be written as the numerator [latex](3)[\/latex] to the power of [latex]3[\/latex], and the denominator [latex](4)[\/latex] to the power of [latex]3[\/latex].\r\n\r\nSimilarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.\r\n<table id=\"eip-id1168469592930\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(\\Large\\frac{x}{y}\\normalsize\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>This means<\/td>\r\n<td>[latex]\\Large\\frac{x}{y}\\normalsize\\cdot\\Large\\frac{x}{y}\\normalsize\\cdot\r\n\r\n\\Large\\frac{x}{y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the fractions.<\/td>\r\n<td>[latex]\\Large\\frac{x\\cdot x\\cdot x}{y\\cdot y\\cdot y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write with exponents.<\/td>\r\n<td>[latex]\\Large\\frac{{x}^{3}}{{y}^{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that the exponent applies to both the numerator and the denominator.\u00a0\u00a0This leads to the Quotient to a Power Property for Exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>Quotient to a Power Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]m[\/latex] is a counting number, then\r\n\r\n[latex]{\\left(\\Large\\frac{a}{b}\\normalsize\\right)}^{m}=\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex]\r\nTo raise a fraction to a power, raise the numerator and denominator to that power.\r\n\r\n<\/div>\r\nAn example with numbers may help you understand this property:\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(\\Large\\frac{5}{8}\\normalsize\\right)}^{2}[\/latex]\r\n2. [latex]{\\left(\\Large\\frac{x}{3}\\normalsize\\right)}^{4}[\/latex]\r\n3. [latex]{\\left(\\Large\\frac{y}{m}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[reveal-answer q=\"803388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803388\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467475453\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5 over 8 raised to the second power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large(\\frac{5}{8})^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{5^{\\color{red}{2}}}{8^{\\color{red}{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{25}{64}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469801450\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x over 3 raised to the fourth power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large(\\frac{x}{3})^4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{x^{\\color{red}{4}}}{3^{\\color{red}{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{x^4}{81}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468503320\" class=\"unnumbered unstyled\" summary=\"Parentheses y over m raised to the third power is shown. Below that, it says, \">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large(\\frac{y}{m})^3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Raise the numerator and denominator to the third power.<\/td>\r\n<td>[latex]\\Large\\frac{y^{3}}{m^{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146227[\/ohm_question]\r\n\r\n[ohm_question]146891[\/ohm_question]\r\n\r\n[ohm_question]146892[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]\r\n\r\n[reveal-answer q=\"875425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875425\"]Apply the power to each factor individually.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\r\nSeparate into numerical and variable factors.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\r\nSimplify by taking [latex]2[\/latex] to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nFor more examples of how to simplify a quotient raised to a power, watch the following video.\r\n\r\nhttps:\/\/youtu.be\/BoBe31pRxFM\r\nIn the following video you will be shown examples of simplifying quotients that are raised to a power.\r\n\r\nhttps:\/\/youtu.be\/ZbxgDRV35dE\r\n<h2>Contribute!<\/h2><div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div><a href=\"https:\/\/docs.google.com\/document\/d\/1Pym3VH1FkoBqqtPvmNwXjObebIImixBNfztYowhJYkg\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions using the Quotient Property of Exponents<\/li>\n<\/ul>\n<\/div>\n<h2>Simplify Expressions Using the Quotient Property of Exponents<\/h2>\n<p>Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.<\/p>\n<div class=\"textbox shaded\">\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\n<p>If [latex]a\\text{ and }b[\/latex] are real numbers and [latex]m\\text{ and }n[\/latex] are whole numbers, then<\/p>\n<p>[latex]\\begin{array}{cccc}\\text{Product Property}\\hfill & & & \\hfill {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\text{Power Property}\\hfill & & & \\hfill {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\text{Product to a Power}\\hfill & & & \\hfill {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<p>Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions\u2014which are also quotients.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then<\/p>\n<p>[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]<\/p>\n<\/div>\n<p>As before, we&#8217;ll try to discover a property by looking at some examples.<\/p>\n<p>Let\u2019s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\n<p>You can rewrite the expression as: [latex]\\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of [latex]4[\/latex] in the numerator and denominator: [latex]\\displaystyle[\/latex]<\/p>\n<p>Finally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, [latex]3[\/latex], is the difference between the two exponents in the original expression, [latex]5[\/latex] and [latex]2[\/latex].<\/p>\n<p>So,\u00a0[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].<\/p>\n<p>Now, let&#8217;s consider an example in which the base is the variable [latex]x[\/latex].<\/p>\n<p>[latex]\\begin{array}{cccccccccc}\\text{Consider}\\hfill & & & \\hfill {\\Large\\frac{{x}^{5}}{{x}^{2}}}\\hfill & & & \\text{and}\\hfill & & & \\hfill {\\Large\\frac{{x}^{2}}{{x}^{3}}}\\hfill \\\\ \\text{What do they mean?}\\hfill & & & \\hfill {\\Large\\frac{x\\cdot x\\cdot x\\cdot x\\cdot x}{x\\cdot x}}\\hfill & & & & & & \\hfill {\\Large\\frac{x\\cdot x}{x\\cdot x\\cdot x}}\\hfill \\\\ \\text{Use the Equivalent Fractions Property.}\\hfill & & & \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot x\\cdot x\\cdot x}{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}\\hfill & & & & & & \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}{\\overline{)x}\\cdot \\overline{)x}\\cdot x}\\hfill \\\\ \\text{Simplify.}\\hfill & & & \\hfill {x}^{3}\\hfill & & & & & & \\hfill {\\Large\\frac{1}{x}}\\hfill \\end{array}[\/latex]<\/p>\n<p>Notice that in each case the bases were the same and we subtracted the exponents.\u00a0\u00a0So, to divide two exponential terms with the same base, subtract the exponents.<\/p>\n<ul id=\"fs-id916657\">\n<li>When the larger exponent was in the numerator, we were left with factors in the numerator and [latex]1[\/latex] in the denominator, which we simplified.<\/li>\n<li>When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[\/latex] in the numerator, which could not be simplified.<\/li>\n<\/ul>\n<p>We write:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}\\frac{{x}^{5}}{{x}^{2}}\\hfill & & & & \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ {x}^{5 - 2}\\hfill & & & & \\hfill \\frac{1}{{x}^{3 - 2}}\\hfill \\\\ {x}^{3}\\hfill & & & & \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then<\/p>\n<p>[latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n\\text{ and }{\\Large\\frac{{a}^{m}}{{a}^{n}}}={\\Large\\frac{1}{{a}^{n-m}}},n>m[\/latex]<\/p>\n<\/div>\n<p>A couple of examples with numbers may help to verify this property.<\/p>\n<p>[latex]\\begin{array}{cccc}\\frac{{3}^{4}}{{3}^{2}}\\stackrel{?}{=}{3}^{4 - 2}\\hfill & & & \\hfill \\frac{{5}^{2}}{{5}^{3}}\\stackrel{?}{=}\\frac{1}{{5}^{3 - 2}}\\hfill \\\\ \\frac{81}{9}\\stackrel{?}{=}{3}^{2}\\hfill & & & \\hfill \\frac{25}{125}\\stackrel{?}{=}\\frac{1}{{5}^{1}}\\hfill \\\\ 9=9 \\hfill & & & \\hfill \\frac{1}{5}=\\frac{1}{5}\\hfill \\end{array}[\/latex]<\/p>\n<p>When we work with numbers and the exponent is less than or equal to [latex]3[\/latex], we will apply the exponent. When the exponent is greater than [latex]3[\/latex] , we leave the answer in exponential form.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]<br \/>\n2. [latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/p>\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168469856047\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 10th over x to the 8th. Beside that is written\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since 10 &gt; 8, there are more factors of [latex]x[\/latex] in the numerator.<\/td>\n<td>[latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]m>n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex] .<\/td>\n<td>[latex]{x}^{\\color{red}{10-8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since 9 &gt; 2, there are more factors of 2 in the numerator.<\/td>\n<td>[latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]m>n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex].<\/td>\n<td>[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146219\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146219&theme=oea&iframe_resize_id=ohm146219&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]<br \/>\n2. [latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q738923\">Show Solution<\/span><\/p>\n<div id=\"q738923\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168469768330\" class=\"unnumbered unstyled\" summary=\"The first line shows b to the 10th over b to the 15th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">1.<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15.34375px;\">\n<td style=\"height: 15.34375px;\">Since [latex]15>10[\/latex], there are more factors of [latex]b[\/latex] in the denominator.<\/td>\n<td style=\"height: 15.34375px;\">[latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Use the quotient property with [latex]n>m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td style=\"height: 15px;\">[latex]\\Large\\frac{\\color{red}{1}}{{b}^{\\color{red}{15-10}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]\\Large\\frac{1}{{b}^{5}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469585119\" class=\"unnumbered unstyled\" summary=\"The first line shows 3 to the 3rd over 3 to the 5th. Beside that is written\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since [latex]5>3[\/latex], there are more factors of [latex]3[\/latex] in the denominator.<\/td>\n<td>[latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]n>m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{1}}{{3}^{\\color{red}{5-3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{1}{{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Apply the exponent.<\/td>\n<td>[latex]\\Large\\frac{1}{9}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146220\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146220&theme=oea&iframe_resize_id=ohm146220&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now let&#8217;s see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]<br \/>\n2. [latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q903400\">Show Solution<\/span><\/p>\n<div id=\"q903400\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468311860\" class=\"unnumbered unstyled\" summary=\"The first line shows a to the 5th over a to the 9th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15.2344px;\">\n<td style=\"height: 15.2344px;\">1.<\/td>\n<td style=\"height: 15.2344px;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Since [latex]9>5[\/latex], there are more [latex]a[\/latex] &#8216;s in the denominator and so we will end up with factors in the denominator.<\/td>\n<td style=\"height: 30px;\">[latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Use the Quotient Property for [latex]n>m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td style=\"height: 30px;\">[latex]\\Large\\frac{\\color{red}{1}}{{a}^{\\color{red}{9-5}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]\\Large\\frac{1}{{a}^{4}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468257607\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">2.<\/td>\n<td style=\"height: 15px;\"><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td style=\"height: 30px;\">Notice there are more factors of [latex]x[\/latex] in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.<\/td>\n<td style=\"height: 30px;\">[latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.28125px;\">\n<td style=\"height: 15.28125px;\">Use the Quotient Property for [latex]m>n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{n-m}[\/latex].<\/td>\n<td style=\"height: 15.28125px;\">[latex]{x}^{\\color{red}{11-7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Simplify.<\/td>\n<td style=\"height: 15px;\">[latex]{x}^{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146889\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146889&theme=oea&iframe_resize_id=ohm146889&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23604\">Show Solution<\/span><\/p>\n<div id=\"q23604\" class=\"hidden-answer\" style=\"display: none\">Separate into numerical and variable factors.[latex]\\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\n<p>Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.<\/p>\n<p>[latex]\\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\n<p>Answer<br \/>\n[latex]\\frac{12{{x}^{4}}}{2x}=6{{x}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Jmf-CPhm3XM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplify Quotients Raised to a Power<\/h2>\n<p>Now we will look at an example that will lead us to the Quotient to a Power Property.<\/p>\n<p>Now let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex]\\displaystyle \\frac{3}{4}[\/latex] and raise it to the [latex]3<sup>rd<\/sup>[\/latex]power.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\n<p>You can see that raising the quotient to the power of [latex]3[\/latex] can also be written as the numerator [latex](3)[\/latex] to the power of [latex]3[\/latex], and the denominator [latex](4)[\/latex] to the power of [latex]3[\/latex].<\/p>\n<p>Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.<\/p>\n<table id=\"eip-id1168469592930\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(\\Large\\frac{x}{y}\\normalsize\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>This means<\/td>\n<td>[latex]\\Large\\frac{x}{y}\\normalsize\\cdot\\Large\\frac{x}{y}\\normalsize\\cdot    \\Large\\frac{x}{y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the fractions.<\/td>\n<td>[latex]\\Large\\frac{x\\cdot x\\cdot x}{y\\cdot y\\cdot y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write with exponents.<\/td>\n<td>[latex]\\Large\\frac{{x}^{3}}{{y}^{3}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the exponent applies to both the numerator and the denominator.\u00a0\u00a0This leads to the Quotient to a Power Property for Exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient to a Power Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]m[\/latex] is a counting number, then<\/p>\n<p>[latex]{\\left(\\Large\\frac{a}{b}\\normalsize\\right)}^{m}=\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex]<br \/>\nTo raise a fraction to a power, raise the numerator and denominator to that power.<\/p>\n<\/div>\n<p>An example with numbers may help you understand this property:<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(\\Large\\frac{5}{8}\\normalsize\\right)}^{2}[\/latex]<br \/>\n2. [latex]{\\left(\\Large\\frac{x}{3}\\normalsize\\right)}^{4}[\/latex]<br \/>\n3. [latex]{\\left(\\Large\\frac{y}{m}\\normalsize\\right)}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803388\">Show Solution<\/span><\/p>\n<div id=\"q803388\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467475453\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5 over 8 raised to the second power. The next line says,\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{5}{8})^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{5^{\\color{red}{2}}}{8^{\\color{red}{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{25}{64}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469801450\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x over 3 raised to the fourth power. The next line says,\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{x}{3})^4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{x^{\\color{red}{4}}}{3^{\\color{red}{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{x^4}{81}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468503320\" class=\"unnumbered unstyled\" summary=\"Parentheses y over m raised to the third power is shown. Below that, it says,\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{y}{m})^3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Raise the numerator and denominator to the third power.<\/td>\n<td>[latex]\\Large\\frac{y^{3}}{m^{3}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146227\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146227&theme=oea&iframe_resize_id=ohm146227&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146891\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146891&theme=oea&iframe_resize_id=ohm146891&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146892\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146892&theme=oea&iframe_resize_id=ohm146892&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875425\">Show Solution<\/span><\/p>\n<div id=\"q875425\" class=\"hidden-answer\" style=\"display: none\">Apply the power to each factor individually.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\n<p>Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\n<p>Simplify by taking [latex]2[\/latex] to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>For more examples of how to simplify a quotient raised to a power, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Expressions Using Exponent Rules (Power of a Quotient)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BoBe31pRxFM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nIn the following video you will be shown examples of simplifying quotients that are raised to a power.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZbxgDRV35dE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a href=\"https:\/\/docs.google.com\/document\/d\/1Pym3VH1FkoBqqtPvmNwXjObebIImixBNfztYowhJYkg\" target=\"_blank\" style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\">Improve this page<\/a><a style=\"margin-left: 16px;\" target=\"_blank\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\">Learn More<\/a><\/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-10850\">\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>Question ID: 146892, 146891, 146227, 146222, 146223, 146890, 146221, 146889, 146220, 146219 . <strong>Authored by<\/strong>: Alyson Day. <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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Jmf-CPhm3XM\">https:\/\/youtu.be\/Jmf-CPhm3XM<\/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>Ex 3: Exponent Properties (Zero Exponent). <strong>Authored by<\/strong>: Lumen Learning. <strong>Provided by<\/strong>: q. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zQJy1aBm1dQ\">https:\/\/youtu.be\/zQJy1aBm1dQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Expressions Using Exponent Rules (Power of a Quotient). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BoBe31pRxFM\">https:\/\/youtu.be\/BoBe31pRxFM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: 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