{"id":1651,"date":"2016-06-22T13:21:26","date_gmt":"2016-06-22T13:21:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1651"},"modified":"2019-07-24T21:27:25","modified_gmt":"2019-07-24T21:27:25","slug":"read-or-watch-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-or-watch-rational-exponents\/","title":{"raw":"Radical Expressions and Rational Exponents","rendered":"Radical Expressions and Rational Exponents"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Convert radicals to expressions with rational exponents<\/li>\r\n \t<li>Convert expressions with rational exponents to their radical equivalent<\/li>\r\n<\/ul>\r\n<\/div>\r\nSquare roots are most often written using a radical sign, like this, [latex] \\sqrt{4}[\/latex]. But there is another way to represent them. You can use rational exponents instead of a radical. A <strong>rational exponent<\/strong> is an exponent that is a fraction. For example, [latex] \\sqrt{4}[\/latex] can be written as [latex] {{4}^{\\tfrac{1}{2}}}[\/latex].\r\n\r\nHaving difficulty imagining a number being raised to a rational power? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions.\r\n\r\n<strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \\sqrt{16}[\/latex], to quite complicated, as in [latex] \\sqrt[3]{250{{x}^{4}}y}[\/latex].\r\n<h2>Write an Expression with a Rational Exponent as a Radical<\/h2>\r\nRadicals and fractional exponents are alternate ways of expressing the same thing. \u00a0In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.\r\n<table style=\"width: 30%;\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center;\">Radical Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Exponent Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Principal Root<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt{16}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{16}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt{25}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{25}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt{100}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{100}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet us look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number\u00a0[latex]3[\/latex].\r\n<table style=\"width: 30%;\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center;\">Radical Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Exponent Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Principal Root<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{8}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{8}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{125}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{125}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{1000}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{1000}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese examples help us model a relationship between radicals and rational exponents: namely, that the <i>n<\/i>th root of a number can be written as either [latex] \\sqrt[n]{x}[\/latex] or [latex] {{x}^{\\frac{1}{n}}}[\/latex].\r\n<table style=\"width: 30%;\">\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center;\">Radical Form<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Exponent Form<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt{x}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{x}^{\\tfrac{1}{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{x}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{x}^{\\tfrac{1}{3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[4]{x}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{x}^{\\tfrac{1}{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">\u2026<\/td>\r\n<td style=\"text-align: center;\">\u2026<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[n]{x}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex] {{x}^{\\tfrac{1}{n}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of [latex] \\frac{1}{2}[\/latex] translates to the square root, an exponent of [latex] \\frac{1}{5}[\/latex] translates to the fifth root or [latex] \\sqrt[5]{\\quad}[\/latex], and [latex] \\frac{1}{8}[\/latex] translates to the eighth root or [latex] \\sqrt[8]{\\quad}[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nExpress [latex] {{(2x)}^{^{\\frac{1}{3}}}}[\/latex] in radical form.\r\n\r\n[reveal-answer q=\"581351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"581351\"]\r\n\r\nRewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.\r\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{2x} [\/latex]<\/p>\r\nThe parentheses in [latex] {{\\left( 2x \\right)}^{\\frac{1}{3}}}[\/latex] indicate that the exponent refers to everything within the parentheses.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference\u2014there are no parentheses! Look what happens.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nExpress [latex] 2{{x}^{^{\\frac{1}{3}}}}[\/latex] in radical form.\r\n\r\n[reveal-answer q=\"236347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236347\"]\r\n\r\nRewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[3]{x}[\/latex]<\/p>\r\nThe exponent refers only to the part of the expression immediately to the left of the exponent, in this case <i>x, <\/i>but not the\u00a0[latex]2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Write a Radical Expression as an Expression with a Rational Exponent<\/h2>\r\n[caption id=\"attachment_3123\" align=\"alignright\" width=\"141\"]<img class=\" wp-image-3123\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/26174543\/Screen-Shot-2016-07-26-at-10.44.01-AM-300x291.png\" alt=\"Person sitting on the ground with one leg arched behind them and one leg curved in front of them.\" width=\"141\" height=\"137\" \/> Flexibility[\/caption]\r\n\r\n&nbsp;\r\n\r\nWe can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows us to have flexibility in solving and simplifying them. It is like having a thesaurus when you write. You want to have options for expressing yourself!\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite [latex] \\sqrt[4]{81}[\/latex] as an expression with a rational exponent.\r\n[reveal-answer q=\"612743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612743\"]\r\n\r\nThe radical form [latex] \\Large\\sqrt[4]{{\\,\\,\\,\\,}}[\/latex] can be rewritten as the exponent [latex] \\frac{1}{4}[\/latex]. Remove the radical and place the exponent next to the base.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nExpress [latex] 4\\sqrt[3]{xy}[\/latex] with rational exponents.\r\n\r\n[reveal-answer q=\"527560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"527560\"]\r\n\r\nRewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is\u00a0[latex]3[\/latex], so the rational exponent will be [latex] \\frac{1}{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex] 4{{(xy)}^{\\frac{1}{3}}}[\/latex]<\/p>\r\nSince\u00a0[latex]4[\/latex] is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Rational Exponents Whose Numerator is Not Equal to One<\/h2>\r\nAll of the numerators for the fractional exponents in the examples above were\u00a0[latex]1[\/latex]. You can use fractional exponents that have numerators other than\u00a0[latex]1[\/latex] to express roots, as shown below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>\r\n<p style=\"text-align: center;\">Radical<\/p>\r\n<\/th>\r\n<th>\r\n<p style=\"text-align: center;\">Exponent<\/p>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt{9}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{1}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex] \\sqrt[3]{{{9}^{2}}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\\sqrt[4]{9^{3}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{3}{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\\sqrt[5]{9^{2}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">\u2026<\/td>\r\n<td style=\"text-align: center;\">\u2026<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]\\sqrt[n]{9^{x}}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]9^{\\frac{x}{n}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\" wp-image-3198 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/29225734\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png\" alt=\"Screen Shot 2016-07-29 at 3.56.45 PM\" width=\"380\" height=\"227\" \/>\r\n\r\nTo rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root\/ index becomes the denominator.\r\n<div class=\"textbox shaded\">\r\n<h3>Writing Rational\u00a0Exponents<\/h3>\r\nAny radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex].\r\n\r\n<\/div>\r\nThe relationship between [latex] \\sqrt[n]{{{a}^{x}}}[\/latex]and [latex] {{a}^{\\frac{x}{n}}}[\/latex] works for rational exponents that have a numerator of\u00a0[latex]1[\/latex] as well. For example, the radical [latex] \\sqrt[3]{8}[\/latex] can also be written as [latex] \\sqrt[3]{{{8}^{1}}}[\/latex], since any number remains the same value if it is raised to the first power. You can now see where the numerator of\u00a0[latex]1[\/latex] comes from in the equivalent form of [latex] {{8}^{\\frac{1}{3}}}[\/latex].\r\n\r\nIn the next example, we practice writing radicals with rational exponents where the numerator is not equal to one.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the radicals using a rational exponent, then simplify your result.\r\n<ol>\r\n \t<li>[latex]\\sqrt[3]{{{a}^{6}}}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[12]{16^3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"898415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"898415\"]\r\n\r\n1.[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=3,\\text{ and }x=6[\/latex], therefore\r\n\r\n[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{\\frac{6}{3}}}[\/latex]\r\n\r\nSimplify the exponent.\r\n\r\n[latex]{{a}^{\\frac{6}{3}}}={{a}^{2}}[\/latex]\r\n\r\n2.\u00a0[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=12,\\text{ and }x=3[\/latex], therefore\r\n<p style=\"text-align: center;\">[latex]\\sqrt[12]{16^3}={16}^{\\frac{3}{12}}={16}^{\\frac{1}{4}}[\/latex]<\/p>\r\nSimplify the expression using rules for exponents.\r\n\r\n[latex]{16}^{\\frac{1}{4}}=2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions and as we learn how to solve radical equations. Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how\u00a0the numerator and denominator of the exponent are\u00a0the exponent of a radicand and index of a radical.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the expressions\u00a0using a radical.\r\n<ol>\r\n \t<li>[latex]{x}^{\\frac{2}{3}}[\/latex]<\/li>\r\n \t<li>[latex]{5}^{\\frac{4}{7}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"200228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200228\"]\r\n<ol>\r\n \t<li>[latex]{x}^{\\frac{2}{3}}[\/latex], the numerator is\u00a0[latex]2[\/latex] and the denominator is\u00a0[latex]3[\/latex], therefore we will have the third root of x squared, [latex]\\sqrt[3]{x^2}[\/latex]<\/li>\r\n \t<li>[latex]{5}^{\\frac{4}{7}}[\/latex], the numerator is\u00a0[latex]4[\/latex] and the denominator is\u00a0[latex]7[\/latex], so we will have the seventh root of\u00a0[latex]5[\/latex] raised to the fourth power. [latex]\\sqrt[7]{5^4}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the following video, we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions.\r\n\r\nhttps:\/\/youtu.be\/5cWkVrANBWA\r\n\r\nWe will use this notation later, so come back for practice if you forget how\u00a0to write a radical with a rational exponent.\r\n<h2>Summary<\/h2>\r\nAny radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex] can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex]. Rewriting radicals using fractional exponents can be useful when simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Convert radicals to expressions with rational exponents<\/li>\n<li>Convert expressions with rational exponents to their radical equivalent<\/li>\n<\/ul>\n<\/div>\n<p>Square roots are most often written using a radical sign, like this, [latex]\\sqrt{4}[\/latex]. But there is another way to represent them. You can use rational exponents instead of a radical. A <strong>rational exponent<\/strong> is an exponent that is a fraction. For example, [latex]\\sqrt{4}[\/latex] can be written as [latex]{{4}^{\\tfrac{1}{2}}}[\/latex].<\/p>\n<p>Having difficulty imagining a number being raised to a rational power? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions.<\/p>\n<p><strong>Radical expressions<\/strong> are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex]\\sqrt{16}[\/latex], to quite complicated, as in [latex]\\sqrt[3]{250{{x}^{4}}y}[\/latex].<\/p>\n<h2>Write an Expression with a Rational Exponent as a Radical<\/h2>\n<p>Radicals and fractional exponents are alternate ways of expressing the same thing. \u00a0In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.<\/p>\n<table style=\"width: 30%;\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\">Radical Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Exponent Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Principal Root<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{16}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{16}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{25}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{25}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{100}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{100}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let us look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number\u00a0[latex]3[\/latex].<\/p>\n<table style=\"width: 30%;\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\">Radical Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Exponent Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Principal Root<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{8}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{8}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{125}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{125}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{1000}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{1000}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These examples help us model a relationship between radicals and rational exponents: namely, that the <i>n<\/i>th root of a number can be written as either [latex]\\sqrt[n]{x}[\/latex] or [latex]{{x}^{\\frac{1}{n}}}[\/latex].<\/p>\n<table style=\"width: 30%;\">\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\">Radical Form<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Exponent Form<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{x}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{x}^{\\tfrac{1}{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{x}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{x}^{\\tfrac{1}{3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[4]{x}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{x}^{\\tfrac{1}{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\u2026<\/td>\n<td style=\"text-align: center;\">\u2026<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[n]{x}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]{{x}^{\\tfrac{1}{n}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of [latex]\\frac{1}{2}[\/latex] translates to the square root, an exponent of [latex]\\frac{1}{5}[\/latex] translates to the fifth root or [latex]\\sqrt[5]{\\quad}[\/latex], and [latex]\\frac{1}{8}[\/latex] translates to the eighth root or [latex]\\sqrt[8]{\\quad}[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Express [latex]{{(2x)}^{^{\\frac{1}{3}}}}[\/latex] in radical form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q581351\">Show Solution<\/span><\/p>\n<div id=\"q581351\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{2x}[\/latex]<\/p>\n<p>The parentheses in [latex]{{\\left( 2x \\right)}^{\\frac{1}{3}}}[\/latex] indicate that the exponent refers to everything within the parentheses.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference\u2014there are no parentheses! Look what happens.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Express [latex]2{{x}^{^{\\frac{1}{3}}}}[\/latex] in radical form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236347\">Show Solution<\/span><\/p>\n<div id=\"q236347\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[3]{x}[\/latex]<\/p>\n<p>The exponent refers only to the part of the expression immediately to the left of the exponent, in this case <i>x, <\/i>but not the\u00a0[latex]2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Write a Radical Expression as an Expression with a Rational Exponent<\/h2>\n<div id=\"attachment_3123\" style=\"width: 151px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3123\" class=\"wp-image-3123\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/26174543\/Screen-Shot-2016-07-26-at-10.44.01-AM-300x291.png\" alt=\"Person sitting on the ground with one leg arched behind them and one leg curved in front of them.\" width=\"141\" height=\"137\" \/><\/p>\n<p id=\"caption-attachment-3123\" class=\"wp-caption-text\">Flexibility<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>We can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows us to have flexibility in solving and simplifying them. It is like having a thesaurus when you write. You want to have options for expressing yourself!<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write [latex]\\sqrt[4]{81}[\/latex] as an expression with a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q612743\">Show Solution<\/span><\/p>\n<div id=\"q612743\" class=\"hidden-answer\" style=\"display: none\">\n<p>The radical form [latex]\\Large\\sqrt[4]{{\\,\\,\\,\\,}}[\/latex] can be rewritten as the exponent [latex]\\frac{1}{4}[\/latex]. Remove the radical and place the exponent next to the base.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Express [latex]4\\sqrt[3]{xy}[\/latex] with rational exponents.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q527560\">Show Solution<\/span><\/p>\n<div id=\"q527560\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is\u00a0[latex]3[\/latex], so the rational exponent will be [latex]\\frac{1}{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]4{{(xy)}^{\\frac{1}{3}}}[\/latex]<\/p>\n<p>Since\u00a0[latex]4[\/latex] is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Rational Exponents Whose Numerator is Not Equal to One<\/h2>\n<p>All of the numerators for the fractional exponents in the examples above were\u00a0[latex]1[\/latex]. You can use fractional exponents that have numerators other than\u00a0[latex]1[\/latex] to express roots, as shown below.<\/p>\n<table>\n<thead>\n<tr>\n<th>\n<p style=\"text-align: center;\">Radical<\/p>\n<\/th>\n<th>\n<p style=\"text-align: center;\">Exponent<\/p>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt{9}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{1}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[3]{{{9}^{2}}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[4]{9^{3}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{3}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[5]{9^{2}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{2}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">\u2026<\/td>\n<td style=\"text-align: center;\">\u2026<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]\\sqrt[n]{9^{x}}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]9^{\\frac{x}{n}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3198 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/29225734\/Screen-Shot-2016-07-29-at-3.56.45-PM-300x179.png\" alt=\"Screen Shot 2016-07-29 at 3.56.45 PM\" width=\"380\" height=\"227\" \/><\/p>\n<p>To rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root\/ index becomes the denominator.<\/p>\n<div class=\"textbox shaded\">\n<h3>Writing Rational\u00a0Exponents<\/h3>\n<p>Any radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex].<\/p>\n<\/div>\n<p>The relationship between [latex]\\sqrt[n]{{{a}^{x}}}[\/latex]and [latex]{{a}^{\\frac{x}{n}}}[\/latex] works for rational exponents that have a numerator of\u00a0[latex]1[\/latex] as well. For example, the radical [latex]\\sqrt[3]{8}[\/latex] can also be written as [latex]\\sqrt[3]{{{8}^{1}}}[\/latex], since any number remains the same value if it is raised to the first power. You can now see where the numerator of\u00a0[latex]1[\/latex] comes from in the equivalent form of [latex]{{8}^{\\frac{1}{3}}}[\/latex].<\/p>\n<p>In the next example, we practice writing radicals with rational exponents where the numerator is not equal to one.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the radicals using a rational exponent, then simplify your result.<\/p>\n<ol>\n<li>[latex]\\sqrt[3]{{{a}^{6}}}[\/latex]<\/li>\n<li>[latex]\\sqrt[12]{16^3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q898415\">Show Solution<\/span><\/p>\n<div id=\"q898415\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=3,\\text{ and }x=6[\/latex], therefore<\/p>\n<p>[latex]\\sqrt[3]{{{a}^{6}}}={{a}^{\\frac{6}{3}}}[\/latex]<\/p>\n<p>Simplify the exponent.<\/p>\n<p>[latex]{{a}^{\\frac{6}{3}}}={{a}^{2}}[\/latex]<\/p>\n<p>2.\u00a0[latex]\\sqrt[n]{a^{x}}[\/latex] can be rewritten as\u00a0[latex]a^{\\frac{x}{n}}[\/latex], so in this case [latex]n=12,\\text{ and }x=3[\/latex], therefore<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[12]{16^3}={16}^{\\frac{3}{12}}={16}^{\\frac{1}{4}}[\/latex]<\/p>\n<p>Simplify the expression using rules for exponents.<\/p>\n<p>[latex]{16}^{\\frac{1}{4}}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions and as we learn how to solve radical equations. Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how\u00a0the numerator and denominator of the exponent are\u00a0the exponent of a radicand and index of a radical.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the expressions\u00a0using a radical.<\/p>\n<ol>\n<li>[latex]{x}^{\\frac{2}{3}}[\/latex]<\/li>\n<li>[latex]{5}^{\\frac{4}{7}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200228\">Show Solution<\/span><\/p>\n<div id=\"q200228\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]{x}^{\\frac{2}{3}}[\/latex], the numerator is\u00a0[latex]2[\/latex] and the denominator is\u00a0[latex]3[\/latex], therefore we will have the third root of x squared, [latex]\\sqrt[3]{x^2}[\/latex]<\/li>\n<li>[latex]{5}^{\\frac{4}{7}}[\/latex], the numerator is\u00a0[latex]4[\/latex] and the denominator is\u00a0[latex]7[\/latex], so we will have the seventh root of\u00a0[latex]5[\/latex] raised to the fourth power. [latex]\\sqrt[7]{5^4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the following video, we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Write Expressions Using Radicals and Rational Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/5cWkVrANBWA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We will use this notation later, so come back for practice if you forget how\u00a0to write a radical with a rational exponent.<\/p>\n<h2>Summary<\/h2>\n<p>Any radical in the form [latex]\\sqrt[n]{a^{x}}[\/latex] can be written using a fractional exponent in the form [latex]a^{\\frac{x}{n}}[\/latex]. Rewriting radicals using fractional exponents can be useful when simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.<\/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-1651\">\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>Write Expressions Using Radicals and Rational Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5cWkVrANBWA\">https:\/\/youtu.be\/5cWkVrANBWA<\/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>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>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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>Precalculus. <strong>Authored by<\/strong>: Abramson, Jay. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em>. <strong>License Terms<\/strong>: Dwonload fro free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Write Expressions Using Radicals and Rational Exponents\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/5cWkVrANBWA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision 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