{"id":51,"date":"2023-06-21T13:22:28","date_gmt":"2023-06-21T13:22:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/nth-roots-and-rational-exponents\/"},"modified":"2023-07-04T04:55:23","modified_gmt":"2023-07-04T04:55:23","slug":"nth-roots-and-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/nth-roots-and-rational-exponents\/","title":{"raw":"\u25aa   Nth Roots and Rational Exponents","rendered":"\u25aa   Nth Roots and Rational Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify Nth roots.<\/li>\r\n \t<li>Write radicals as rational exponents.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: operations on Fractions<\/h3>\r\nWhen simplifying handling nth roots and rational exponents, we often need to perform operations on fractions. It's important to be able to do these operations on the fractions without converting them to decimals. Recall the rules for operations on fractions.\r\n<ul>\r\n \t<li>To multiply fractions, multiply the numerators and place them over the product of the denominators.\r\n<ul>\r\n \t<li>\u00a0[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>To divide fractions, multiply the first by the reciprocal of the second.\r\n<ul>\r\n \t<li>\u00a0[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>To simplify fractions, find common factors in the numerator and denominator that cancel.\r\n<ul>\r\n \t<li>\u00a0[latex]\\dfrac{24}{32}=\\dfrac{2\\cdot2\\cdot2\\cdot3}{2\\cdot2\\cdot2\\cdot2\\cdot2}=\\dfrac{3}{2\\cdot2}=\\dfrac{3}{4}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>To add or subtract fractions, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.\r\n<ul>\r\n \t<li>\u00a0[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Using Rational Roots<\/h2>\r\nAlthough square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.\r\nSuppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8.\r\n\r\nThe <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex]{\\left(-3\\right)}^{5}=-243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex].\r\n\r\nThe principal <em>n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Principal <em>n<\/em>th Root<\/h3>\r\nIf [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying <em>n<\/em>th Roots<\/h3>\r\nSimplify each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\r\n \t<li>[latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"149528\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149528\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\r\n \t<li>First, express the product as a single radical expression. [latex]\\sqrt[4]{4\\text{,}096}=8[\/latex] because [latex]{8}^{4}=4,096[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}\\\\ &amp;\\frac{-\\sqrt[3]{8{x}^{6}}}{\\sqrt[3]{125}} &amp;&amp; \\text{Write as quotient of two radical expressions}. \\\\ &amp;\\frac{-2{x}^{2}}{5} &amp;&amp; \\text{Simplify}. \\\\ \\end{align}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{align}\\\\ &amp;8\\sqrt[4]{3}-2\\sqrt[4]{3} &amp;&amp; \\text{Simplify to get equal radicands}. \\\\ &amp;6\\sqrt[4]{3} &amp;&amp; \\text{Add}. \\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.\r\n<ol>\r\n \t<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\r\n \t<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"15987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15987\"]\r\n<ol>\r\n \t<li>[latex]-6[\/latex]<\/li>\r\n \t<li>[latex]6[\/latex]<\/li>\r\n \t<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2564&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2565&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2567&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2592&amp;theme=oea&amp;iframe_resize_id=mom4[\/embed]\r\n\r\n<\/div>\r\n<h2>Using Rational Exponents<\/h2>\r\nRadical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.\r\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\r\nWe can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an <em>n<\/em>th root. The numerator tells us the power and the denominator tells us the root.\r\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\r\nAll of the properties of exponents that we learned for integer exponents also hold for rational exponents.\r\n<div class=\"textbox\">\r\n<h3>Rational Exponents<\/h3>\r\nRational exponents are another way to express principal <em>n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is\r\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a rational exponent, write the expression as a radical.<\/h3>\r\n<ol>\r\n \t<li>Determine the power by looking at the numerator of the exponent.<\/li>\r\n \t<li>Determine the root by looking at the denominator of the exponent.<\/li>\r\n \t<li>Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing Rational Exponents as Radicals<\/h3>\r\nWrite [latex]{343}^{\\frac{2}{3}}[\/latex] as a radical. Simplify.\r\n\r\n[reveal-answer q=\"878113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"878113\"]\r\nThe 2 tells us the power and the 3 tells us the root.\r\n<p style=\"text-align: center;\">[latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}=\\sqrt[3]{{343}^{2}}[\/latex]<\/p>\r\nWe know that [latex]\\sqrt[3]{343}=7[\/latex] because [latex]{7}^{3}=343[\/latex]. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.\r\n<p style=\"text-align: center;\">[latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}={7}^{2}=49[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.\r\n\r\n[reveal-answer q=\"937831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937831\"]\r\n\r\n[latex]{\\left(\\sqrt{9}\\right)}^{5}={3}^{5}=243[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3415&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing Radicals as Rational Exponents<\/h3>\r\nWrite [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.\r\n\r\n[reveal-answer q=\"183909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"183909\"]\r\n\r\nThe power is 2 and the root is 7, so the rational exponent will be [latex]\\dfrac{2}{7}[\/latex]. We get [latex]\\dfrac{4}{{a}^{\\frac{2}{7}}}[\/latex]. Using properties of exponents, we get [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}=4{a}^{\\frac{-2}{7}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.\r\n\r\n[reveal-answer q=\"522860\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"522860\"]\r\n\r\n[latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch this video to see more examples of how to write a radical with a fractional exponent.\r\n\r\nhttps:\/\/youtu.be\/L5Z_3RrrVjA\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Rational Exponents<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"803060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803060\"]\r\n\r\n1.\r\n[latex]\\begin{align}&amp;30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}&amp;&amp; \\text{Multiply the coefficients}. \\\\ &amp;30{x}^{\\frac{3}{4}+\\frac{1}{5}}&amp;&amp; \\text{Use properties of exponents}. \\\\ &amp;30{x}^{\\frac{19}{20}}&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]\r\n\r\n2.\r\n[latex]\\begin{align}&amp;{\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}&amp;&amp; \\text{Use definition of negative exponents}. \\\\ &amp;\\sqrt{\\frac{9}{16}}&amp;&amp; \\text{Rewrite as a radical}. \\\\ &amp;\\frac{\\sqrt{9}}{\\sqrt{16}}&amp;&amp; \\text{Use the quotient rule}. \\\\ &amp;\\frac{3}{4}&amp;&amp; \\text{Simplify}. \\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/latex]\r\n\r\n[reveal-answer q=\"95703\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95703\"]\r\n\r\n[latex]28{x}^{\\frac{23}{15}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\nSimplify [latex]\\large{\\frac{\\sqrt{y}}{y^\\frac{2}{5}}}[\/latex]\r\n\r\n[reveal-answer q=\"95714\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95714\"]\r\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{2}}-y^{\\frac{2}{5}}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{5}{10}}-y^{\\frac{4}{10}}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{10}}[\/latex] or [latex]\\sqrt[10]{y}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n[ohm_question]59783[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify Nth roots.<\/li>\n<li>Write radicals as rational exponents.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: operations on Fractions<\/h3>\n<p>When simplifying handling nth roots and rational exponents, we often need to perform operations on fractions. It&#8217;s important to be able to do these operations on the fractions without converting them to decimals. Recall the rules for operations on fractions.<\/p>\n<ul>\n<li>To multiply fractions, multiply the numerators and place them over the product of the denominators.\n<ul>\n<li>\u00a0[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>To divide fractions, multiply the first by the reciprocal of the second.\n<ul>\n<li>\u00a0[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>To simplify fractions, find common factors in the numerator and denominator that cancel.\n<ul>\n<li>\u00a0[latex]\\dfrac{24}{32}=\\dfrac{2\\cdot2\\cdot2\\cdot3}{2\\cdot2\\cdot2\\cdot2\\cdot2}=\\dfrac{3}{2\\cdot2}=\\dfrac{3}{4}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>To add or subtract fractions, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.\n<ul>\n<li>\u00a0[latex]\\dfrac{a}{b}\\pm\\dfrac{c}{d} = \\dfrac{ad \\pm bc}{bd}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Using Rational Roots<\/h2>\n<p>Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.<br \/>\nSuppose we know that [latex]{a}^{3}=8[\/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[\/latex], we say that 2 is the cube root of 8.<\/p>\n<p>The <em>n<\/em>th root of [latex]a[\/latex] is a number that, when raised to the <em>n<\/em>th power, gives [latex]a[\/latex]. For example, [latex]-3[\/latex] is the 5th root of [latex]-243[\/latex] because [latex]{\\left(-3\\right)}^{5}=-243[\/latex]. If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex] is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex].<\/p>\n<p>The principal <em>n<\/em>th root of [latex]a[\/latex] is written as [latex]\\sqrt[n]{a}[\/latex], where [latex]n[\/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[\/latex] is called the <strong>index<\/strong> of the radical.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Principal <em>n<\/em>th Root<\/h3>\n<p>If [latex]a[\/latex] is a real number with at least one <em>n<\/em>th root, then the <strong>principal <em>n<\/em>th root<\/strong> of [latex]a[\/latex], written as [latex]\\sqrt[n]{a}[\/latex], is the number with the same sign as [latex]a[\/latex] that, when raised to the <em>n<\/em>th power, equals [latex]a[\/latex]. The <strong>index<\/strong> of the radical is [latex]n[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying <em>n<\/em>th Roots<\/h3>\n<p>Simplify each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\n<li>[latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\n<li>[latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\n<li>[latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149528\">Show Solution<\/span><\/p>\n<div id=\"q149528\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\n<li>First, express the product as a single radical expression. [latex]\\sqrt[4]{4\\text{,}096}=8[\/latex] because [latex]{8}^{4}=4,096[\/latex]<\/li>\n<li>[latex]\\begin{align}\\\\ &\\frac{-\\sqrt[3]{8{x}^{6}}}{\\sqrt[3]{125}} && \\text{Write as quotient of two radical expressions}. \\\\ &\\frac{-2{x}^{2}}{5} && \\text{Simplify}. \\\\ \\end{align}[\/latex]<\/li>\n<li>[latex]\\begin{align}\\\\ &8\\sqrt[4]{3}-2\\sqrt[4]{3} && \\text{Simplify to get equal radicands}. \\\\ &6\\sqrt[4]{3} && \\text{Add}. \\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.<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\n<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\n<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15987\">Show Solution<\/span><\/p>\n<div id=\"q15987\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-6[\/latex]<\/li>\n<li>[latex]6[\/latex]<\/li>\n<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm2564\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2564&#38;theme=oea&#38;iframe_resize_id=ohm2564&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2565\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2565&#38;theme=oea&#38;iframe_resize_id=ohm2565&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2567\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2567&#38;theme=oea&#38;iframe_resize_id=ohm2567&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm2592\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2592&#38;theme=oea&#38;iframe_resize_id=ohm2592&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Using Rational Exponents<\/h2>\n<p>Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\n<p>We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an <em>n<\/em>th root. The numerator tells us the power and the denominator tells us the root.<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\n<p>All of the properties of exponents that we learned for integer exponents also hold for rational exponents.<\/p>\n<div class=\"textbox\">\n<h3>Rational Exponents<\/h3>\n<p>Rational exponents are another way to express principal <em>n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a rational exponent, write the expression as a radical.<\/h3>\n<ol>\n<li>Determine the power by looking at the numerator of the exponent.<\/li>\n<li>Determine the root by looking at the denominator of the exponent.<\/li>\n<li>Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Rational Exponents as Radicals<\/h3>\n<p>Write [latex]{343}^{\\frac{2}{3}}[\/latex] as a radical. Simplify.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q878113\">Show Solution<\/span><\/p>\n<div id=\"q878113\" class=\"hidden-answer\" style=\"display: none\">\nThe 2 tells us the power and the 3 tells us the root.<\/p>\n<p style=\"text-align: center;\">[latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}=\\sqrt[3]{{343}^{2}}[\/latex]<\/p>\n<p>We know that [latex]\\sqrt[3]{343}=7[\/latex] because [latex]{7}^{3}=343[\/latex]. Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.<\/p>\n<p style=\"text-align: center;\">[latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}={7}^{2}=49[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q937831\">Show Solution<\/span><\/p>\n<div id=\"q937831\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{\\left(\\sqrt{9}\\right)}^{5}={3}^{5}=243[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3415\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3415&#38;theme=oea&#38;iframe_resize_id=ohm3415&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Radicals as Rational Exponents<\/h3>\n<p>Write [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q183909\">Show Solution<\/span><\/p>\n<div id=\"q183909\" class=\"hidden-answer\" style=\"display: none\">\n<p>The power is 2 and the root is 7, so the rational exponent will be [latex]\\dfrac{2}{7}[\/latex]. We get [latex]\\dfrac{4}{{a}^{\\frac{2}{7}}}[\/latex]. Using properties of exponents, we get [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}=4{a}^{\\frac{-2}{7}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q522860\">Show Solution<\/span><\/p>\n<div id=\"q522860\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch this video to see more examples of how to write a radical with a fractional exponent.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Write a Radical in Rational Exponent Form\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/L5Z_3RrrVjA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Rational Exponents<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\n<li>[latex]{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803060\">Show Solution<\/span><\/p>\n<div id=\"q803060\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<br \/>\n[latex]\\begin{align}&30{x}^{\\frac{3}{4}}{x}^{\\frac{1}{5}}&& \\text{Multiply the coefficients}. \\\\ &30{x}^{\\frac{3}{4}+\\frac{1}{5}}&& \\text{Use properties of exponents}. \\\\ &30{x}^{\\frac{19}{20}}&& \\text{Simplify}. \\end{align}[\/latex]<\/p>\n<p>2.<br \/>\n[latex]\\begin{align}&{\\left(\\frac{9}{16}\\right)}^{\\frac{1}{2}}&& \\text{Use definition of negative exponents}. \\\\ &\\sqrt{\\frac{9}{16}}&& \\text{Rewrite as a radical}. \\\\ &\\frac{\\sqrt{9}}{\\sqrt{16}}&& \\text{Use the quotient rule}. \\\\ &\\frac{3}{4}&& \\text{Simplify}. \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q95703\">Show Solution<\/span><\/p>\n<div id=\"q95703\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]28{x}^{\\frac{23}{15}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>Simplify [latex]\\large{\\frac{\\sqrt{y}}{y^\\frac{2}{5}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q95714\">Show Solution<\/span><\/p>\n<div id=\"q95714\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{2}}-y^{\\frac{2}{5}}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{5}{10}}-y^{\\frac{4}{10}}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]y^{\\frac{1}{10}}[\/latex] or [latex]\\sqrt[10]{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm59783\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=59783&theme=oea&iframe_resize_id=ohm59783&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-51\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Ex: Write a Radical in Rational Exponent Form. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/L5Z_3RrrVjA\">https:\/\/youtu.be\/L5Z_3RrrVjA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 2564, 2565, 2567, 2592. <strong>Authored by<\/strong>: Greg Langkamp. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 3415. <strong>Authored by<\/strong>: Jessica Reidel. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><li>Question ID 59783. <strong>Authored by<\/strong>: Gary Parker. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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