{"id":626,"date":"2021-08-30T20:40:59","date_gmt":"2021-08-30T20:40:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=626"},"modified":"2021-11-23T00:45:21","modified_gmt":"2021-11-23T00:45:21","slug":"2-4-higher-order-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/2-4-higher-order-roots\/","title":{"raw":"2.4: Higher Order Roots","rendered":"2.4: Higher Order Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find cube roots of perfect cubes<\/li>\r\n \t<li>Define even and odd roots<\/li>\r\n \t<li>Determine odd and even roots of perfect nths<\/li>\r\n \t<li>Use rational exponents to rewrite a root<\/li>\r\n \t<li>Estimate non-perfect roots using a calculator<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Cube root<\/strong>: the 3rd root of a number, so that the [latex]\\text{root}^3=\\text{ number}[\/latex]<\/li>\r\n \t<li><strong>Index<\/strong>: the small number in the v-part of the radical that tells which root to take<\/li>\r\n \t<li><strong>Perfect cube<\/strong>: a number whose cube root is an integer<\/li>\r\n \t<li><strong>Principal nth root<\/strong>: the nth root of a number that has the same sign as the number<\/li>\r\n \t<li><strong>Odd roots<\/strong>: roots that have an index that is an odd number<\/li>\r\n \t<li><strong>Even roots<\/strong>: roots\u00a0that have an index that is an even number<\/li>\r\n \t<li><strong>Rational exponent<\/strong>: an exponent that is a fraction<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Cube Roots<\/h2>\r\nWe know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex], but what if we want to \"undo\" [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.\r\n\r\nWhile square roots are probably the most common radical, we can also find the third root, the fifth root, the [latex]10th[\/latex]\u00a0root, or really any other <i>n<\/i>th root of a number. Just as the square root is a number that, when squared, gives the radicand, the <em><strong>cube root<\/strong><\/em> is a number that, when cubed, gives the radicand.\r\n\r\nThe cube root of a number is written with a small number \u00a0[latex]3[\/latex], called the <em><strong>index<\/strong><\/em>, just above the v-part radical symbol. It looks like [latex] \\sqrt[3\\;]{{\\;}}[\/latex]. This little [latex]3[\/latex] distinguishes cube roots from square roots. For a square root the index is [latex]2[\/latex] but it is such a common root that the index is not written.\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>Caution! Be careful to distinguish between [latex] \\sqrt[\\;3]{n}[\/latex], the cube root of [latex]n[\/latex], and [latex] 3\\sqrt{n}[\/latex], three <i>times<\/i> the <i>square<\/i> root of [latex]n[\/latex]. They may look similar at first, but they lead to much different expressions!\r\n\r\n<\/div>\r\nSuppose we know that [latex]{a number}^{3}=8[\/latex]. We want to find what number raised to the\u00a0[latex]3[\/latex]rd power is equal to\u00a0[latex]8[\/latex]. Since [latex]{2}^{3}=8[\/latex], we say that\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex]. In the next example, we will evaluate the cube roots of some perfect cubes.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate the following:\r\n<ol>\r\n \t<li>[latex] \\sqrt[3\\;]{-8}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[3\\;]{27}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{0}[\/latex]<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n1. We want to find a number whose cube is\u00a0[latex]-8[\/latex]. We know\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex], so maybe we can try\u00a0[latex]-2[\/latex] which gives [latex]-2\\cdot{-2}\\cdot{-2}=-8[\/latex], so the cube root of\u00a0[latex]-8[\/latex] is\u00a0[latex]-2[\/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number.\r\n\r\n2. We want to find a number whose cube is\u00a0[latex]27[\/latex]. [latex]3\\cdot{3}\\cdot{3}=27[\/latex], so the cube root of [latex]27[\/latex] is [latex]3[\/latex].\r\n\r\n3.\u00a0 We want to find a number whose cube is [latex]0[\/latex]. [latex]0\\cdot0\\cdot0=0[\/latex], so [latex]\\sqrt[\\;3]{0}=0[\/latex]. No matter how many times we multiply [latex]0[\/latex] by itself, we will always get\u00a0[latex]0[\/latex].\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nAs we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex] \\sqrt[3\\;]{-8}=-2[\/latex] because [latex] -2\\cdot -2\\cdot -2=-8[\/latex]. Remember, when we are multiplying an odd number of negative numbers, the result is negative! Consider [latex] \\sqrt[3\\;]{{{(-1)}^{3}}}=-1[\/latex].\r\n\r\nWe can also use factoring to simplify cube roots such as [latex] \\sqrt[3\\;]{125}[\/latex]. We read this as \u201cthe third root of [latex]125[\/latex]\u201d or \u201cthe cube root of [latex]125[\/latex].\u201d To simplify this expression, look for a number that, when multiplied three times , equals [latex]125[\/latex]. Let\u2019s factor [latex]125[\/latex] and find that number.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3\\;]{125}[\/latex]\r\n<h4>Solution<\/h4>\r\n[latex]125[\/latex] ends in [latex]5[\/latex], so we know that \u00a0[latex]5[\/latex] is a factor. Expand [latex]125[\/latex] into [latex]5\\cdot25[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3\\;]{5\\cdot 25}[\/latex]<\/p>\r\nFactor [latex]25[\/latex] into [latex]5\\cdot 5[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3\\;]{5\\cdot 5\\cdot 5}[\/latex]<\/p>\r\nThe factors are [latex]5\\cdot5\\cdot5[\/latex], or [latex]5^{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3\\;]{{{5}^{3}}}[\/latex]<\/p>\r\nAnswer:\u00a0 \u00a0 \u00a0 [latex] \\sqrt[3\\;]{125}=5[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nThe prime factors of [latex]125[\/latex] are [latex]5\\cdot5\\cdot5[\/latex], which can be rewritten as [latex]5^{3}[\/latex]. The cube root of a cubed number is the number itself, so [latex] \\sqrt[3\\;]{{{5}^{3}}}=5[\/latex]. We have found the cube root, the three identical factors that when multiplied together give [latex]125[\/latex]. [latex]125[\/latex] is known as a <em><strong>perfect cube<\/strong><\/em> because its cube root is an integer.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nEvaluate:\r\n<ol>\r\n \t<li>[latex]\\sqrt[3\\;]{8}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{-8}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{-125}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{64}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{-1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm314\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm314\"]\r\n<ol>\r\n \t<li>[latex]\\sqrt[3\\;]{8}=2[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{-8}=-2[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt[3\\;]{-125}=5[\/latex]<\/li>\r\n \t<li>[latex]-\\sqrt[3\\;]{64}=-4[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[3\\;]{-1}=-1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">nth Roots<\/span>\r\n\r\nWe learned above that the cube root of a number is written with a small number\u00a0[latex]3[\/latex], which looks like [latex] \\sqrt[3\\;]{a}[\/latex]. This number placed just outside and above the radical symbol and is called the <strong><em>index<\/em>.<\/strong>\r\n\r\nWe can apply the same idea to any exponent and its corresponding root. 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 <em><strong>principal nth root<\/strong><\/em> 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\u00a0[latex]2[\/latex]. In the radical expression, [latex]n[\/latex] is called the <em><strong>index<\/strong><\/em> of the radical.\r\n<div class=\"textbox shaded\">\r\n<h3>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<\/h3>\r\nEvaluate each of the following:\r\n<ol>\r\n \t<li>[latex]\\sqrt[5\\;]{-32}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4\\;]{81}[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[8\\;]{-1}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]\\sqrt[5\\;]{-32}[\/latex] Factoring [latex]32[\/latex] gives [latex]{\\left(-2\\right)}^{5}=-32\\text{ so }\\sqrt[5\\;]{-32}=-2[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt[4\\;]{81}[\/latex]. Factoring can help. We know that [latex]9\\cdot9=81[\/latex] and we can further factor each\u00a0[latex]9[\/latex] into [latex]3\\cdot3[\/latex]. So, [latex]81={3^4}[\/latex] and therefore,\u00a0[latex]\\sqrt[4\\;]{81}=3[\/latex].<\/li>\r\n \t<li>[latex]\\sqrt[8\\;]{-1}[\/latex]. Since we have an\u00a0[latex]8[\/latex]th root with a negative number as the radicand, this root has no real number solutions. In other words, there is no real number that when raised to the power 8 results in a negative number.<\/li>\r\n<\/ol>\r\n<\/div>\r\nThe nth roots of a real number are categorized into two groups: <em><strong>odd roots<\/strong><\/em> when the index is an odd number, and <em><strong>even roots<\/strong><\/em> when the index is an even number. Square roots are even roots, since 2 is an even number. The square root of a negative real number is undefined in the set of real numbers. The same can be said for all even roots: the even root of a negative real number is undefined in the set of real numbers. This is because any real number raised to an even power is always positive. For example, [latex](-3)^4=81[\/latex]; [latex](-2)^6=64[\/latex]; [latex](-1)^126=1[\/latex].\r\n\r\nHowever, odd roots, like the cube root, exist for all real numbers. For example, [latex](-2)^5=-32[\/latex], so [latex]\\sqrt[5\\;]{-32}=-2[\/latex], and since [latex]4^3=64[\/latex], [latex]\\sqrt[3\\;]{64}=4[\/latex]. Odd roots take on the sign of the radicand.\r\n\r\nThe steps to consider when simplifying a radical are outlined below.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplifying a radical<\/h3>\r\nWhen working with exponents and radicals:\r\n<ul>\r\n \t<li>If <i>n<\/i> is odd, [latex] \\sqrt[n\\;]{{{x}^{n}}}=x[\/latex].<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 60px;\">The root takes the sign of the radicand.<\/p>\r\n\r\n<ul>\r\n \t<li>If <i>n<\/i> is even, [latex] \\sqrt[n\\;]{{{x}^{n}}}=\\large | \\normalsize x \\large |[\/latex]. (The absolute value accounts for the fact that if [latex]x[\/latex] is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 60px;\">The root is positive and the <em>n<\/em>th root of a negative radicand is undefined in the set of real numbers.<\/p>\r\n\r\n<\/div>\r\nThe video shows more examples of how to evaluate <em>n<\/em>th roots.\r\n\r\nhttps:\/\/youtu.be\/vA2DkcUSRSk\r\n\r\nWe can find the odd root of a negative number, but we cannot find the even root of a negative number in the set of real numbers. This means we can evaluate the radicals [latex] \\sqrt[3\\;]{-81},\\ \\sqrt[5\\;]{-64}[\/latex], and [latex] \\sqrt[7\\;]{-2187}[\/latex] because they all have an odd numbered index, but we cannot evaluate the radicals\u00a0 [latex] \\sqrt[{}]{-100},\\ \\sqrt[4\\;]{-16}[\/latex], or [latex] \\sqrt[6\\;]{-2,500}[\/latex] because they all have an even numbered index.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]94221[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Rational Exponents<\/h2>\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. We can use rational exponents instead of a radical. A <em><strong>rational exponent<\/strong><\/em> is an exponent that is a fraction. For example, [latex] \\sqrt{4}[\/latex] can be written as [latex] {{4}^{\\small\\frac{1}{2}}}[\/latex].\r\n\r\nRadicals and rational 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\nUse the example below to familiarize yourself with the different ways to write square roots.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFill in the missing cells in the table.\r\n<table style=\"width: 50%;\">\r\n<thead>\r\n<tr>\r\n<th>Exponent Form<\/th>\r\n<th>Root Form<\/th>\r\n<th>Root of a Square<\/th>\r\n<th>Simplified<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] {{36}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\sqrt{81}[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>[latex] \\sqrt{{{12}^{2}}}[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"990781\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"990781\"]\r\n<table style=\"width: 50%;\">\r\n<thead>\r\n<tr>\r\n<th>Exponent Form<\/th>\r\n<th>Root Form<\/th>\r\n<th>Root of a Square<\/th>\r\n<th>Simplified<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] {{36}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{36}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{6}^{2}}}[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] {{81}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{81}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{{{9}^{2}}}[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] {{144}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>[latex] \\sqrt{144}[\/latex]<\/td>\r\n<td>[latex]\\sqrt{{{12}^{2}}}[\/latex]<\/td>\r\n<td>[latex]12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show another example of filling in a table to connect the different notation used for roots.\r\n\r\nhttps:\/\/youtu.be\/eGJgmo2CpN4\r\n\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\\;]{x}[\/latex], and [latex] \\frac{1}{8}[\/latex] translates to the eighth root or [latex] \\sqrt[8\\;]{x}[\/latex].\r\n\r\nWhen converting from radical to rational exponent notation, the index of the root becomes the denominator of the exponent. If we start with a square root, we will have an exponent of [latex]\\frac{1}{2}[\/latex] on the radicand. On the other hand, if we start with an exponent of [latex]\\frac{1}{3}[\/latex] we will use a cube root. The following statement summarizes this idea.\r\n<div class=\"textbox shaded\">\r\n<h3>Writing Fractional Exponents<\/h3>\r\nAny radical in the form [latex]\\sqrt[n\\;]{a}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{1}{n}}[\/latex].\r\n\r\n<\/div>\r\n<h2>Simplifying Nth Roots on a Calculator<\/h2>\r\nIn the same way we can find a square root on a calculator, we can find a higher root in the calculator to estimate what the root will be. For example, consider the [latex]\\sqrt[\\;5\\;]{250}[\/latex]. The calculator, depending on your type of calculator, has two different function keys that will take <em>n<\/em>th roots. The first is the key [latex]\\sqrt[y\\;]{x}[\/latex], the second is converting the root to a rational exponent and using the exponent key on the calculator.\r\n\r\nOn the free online scientific calculator from Desmos, we find [latex]\\sqrt[7\\;]{128}[\/latex] by clicking on the nth root button, followed by the index 7, followed by the radicand, 150 after clicking under the radical in the display. The answer shows on the right side of the display and rounds to three decimal places as 2.046.\r\n\r\nWe could get the same answer by writing\u00a0[latex]\\sqrt[7\\;]{128}[\/latex] as [latex]128^\\frac{1}{7}[\/latex] and clicking 128 then the exponent button [latex]a^b[\/latex] followed by the exponent [latex]1\\div 7[\/latex]. As we can see, we get the same answer.\r\n\r\n[caption id=\"attachment_1790\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-1790 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/08\/23000604\/Desmos-nth-roots-1024x621.png\" alt=\"calculator image\" width=\"1024\" height=\"621\" \/> Free Desmos calculator available at desmos.com[\/caption]\r\n\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nUse a calculator to evaluate the roots to 3 decimal places:\r\n\r\n1. [latex]\\sqrt[\\;6\\;]{768}[\/latex]\r\n\r\n2.\u00a0[latex]\\sqrt[\\;4\\;]{321}[\/latex]\r\n\r\n3.\u00a0[latex]\\sqrt[\\;3\\;]{34}[\/latex]\r\n\r\n4.\u00a0[latex]\\sqrt[\\;7\\;]{-7000}[\/latex]\r\n\r\n5.\u00a0[latex]\\sqrt[\\;5\\;]{-96}[\/latex]\r\n\r\n[reveal-answer q=\"hjm655\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm655\"]\r\n<ol>\r\n \t<li>[latex]3.026[\/latex]<\/li>\r\n \t<li>[latex]4.233[\/latex]<\/li>\r\n \t<li>[latex]3.240[\/latex]<\/li>\r\n \t<li>[latex]-3.542[\/latex]<\/li>\r\n \t<li>[latex]-2.491[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find cube roots of perfect cubes<\/li>\n<li>Define even and odd roots<\/li>\n<li>Determine odd and even roots of perfect nths<\/li>\n<li>Use rational exponents to rewrite a root<\/li>\n<li>Estimate non-perfect roots using a calculator<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Cube root<\/strong>: the 3rd root of a number, so that the [latex]\\text{root}^3=\\text{ number}[\/latex]<\/li>\n<li><strong>Index<\/strong>: the small number in the v-part of the radical that tells which root to take<\/li>\n<li><strong>Perfect cube<\/strong>: a number whose cube root is an integer<\/li>\n<li><strong>Principal nth root<\/strong>: the nth root of a number that has the same sign as the number<\/li>\n<li><strong>Odd roots<\/strong>: roots that have an index that is an odd number<\/li>\n<li><strong>Even roots<\/strong>: roots\u00a0that have an index that is an even number<\/li>\n<li><strong>Rational exponent<\/strong>: an exponent that is a fraction<\/li>\n<\/ul>\n<\/div>\n<h2>Cube Roots<\/h2>\n<p>We know that [latex]5^2=25, \\text{ and }\\sqrt{25}=5[\/latex], but what if we want to &#8220;undo&#8221; [latex]5^3=125, \\text{ or }5^4=625[\/latex]? We can use higher order roots to answer these questions.<\/p>\n<p>While square roots are probably the most common radical, we can also find the third root, the fifth root, the [latex]10th[\/latex]\u00a0root, or really any other <i>n<\/i>th root of a number. Just as the square root is a number that, when squared, gives the radicand, the <em><strong>cube root<\/strong><\/em> is a number that, when cubed, gives the radicand.<\/p>\n<p>The cube root of a number is written with a small number \u00a0[latex]3[\/latex], called the <em><strong>index<\/strong><\/em>, just above the v-part radical symbol. It looks like [latex]\\sqrt[3\\;]{{\\;}}[\/latex]. This little [latex]3[\/latex] distinguishes cube roots from square roots. For a square root the index is [latex]2[\/latex] but it is such a common root that the index is not written.<\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"50\" height=\"44\" \/>Caution! Be careful to distinguish between [latex]\\sqrt[\\;3]{n}[\/latex], the cube root of [latex]n[\/latex], and [latex]3\\sqrt{n}[\/latex], three <i>times<\/i> the <i>square<\/i> root of [latex]n[\/latex]. They may look similar at first, but they lead to much different expressions!<\/p>\n<\/div>\n<p>Suppose we know that [latex]{a number}^{3}=8[\/latex]. We want to find what number raised to the\u00a0[latex]3[\/latex]rd power is equal to\u00a0[latex]8[\/latex]. Since [latex]{2}^{3}=8[\/latex], we say that\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex]. In the next example, we will evaluate the cube roots of some perfect cubes.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[3\\;]{-8}[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{27}[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{0}[\/latex]<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>1. We want to find a number whose cube is\u00a0[latex]-8[\/latex]. We know\u00a0[latex]2[\/latex] is the cube root of\u00a0[latex]8[\/latex], so maybe we can try\u00a0[latex]-2[\/latex] which gives [latex]-2\\cdot{-2}\\cdot{-2}=-8[\/latex], so the cube root of\u00a0[latex]-8[\/latex] is\u00a0[latex]-2[\/latex]. This is different from square roots because multiplying three negative numbers together results in a negative number.<\/p>\n<p>2. We want to find a number whose cube is\u00a0[latex]27[\/latex]. [latex]3\\cdot{3}\\cdot{3}=27[\/latex], so the cube root of [latex]27[\/latex] is [latex]3[\/latex].<\/p>\n<p>3.\u00a0 We want to find a number whose cube is [latex]0[\/latex]. [latex]0\\cdot0\\cdot0=0[\/latex], so [latex]\\sqrt[\\;3]{0}=0[\/latex]. No matter how many times we multiply [latex]0[\/latex] by itself, we will always get\u00a0[latex]0[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>As we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]\u22128[\/latex]? [latex]\\sqrt[3\\;]{-8}=-2[\/latex] because [latex]-2\\cdot -2\\cdot -2=-8[\/latex]. Remember, when we are multiplying an odd number of negative numbers, the result is negative! Consider [latex]\\sqrt[3\\;]{{{(-1)}^{3}}}=-1[\/latex].<\/p>\n<p>We can also use factoring to simplify cube roots such as [latex]\\sqrt[3\\;]{125}[\/latex]. We read this as \u201cthe third root of [latex]125[\/latex]\u201d or \u201cthe cube root of [latex]125[\/latex].\u201d To simplify this expression, look for a number that, when multiplied three times , equals [latex]125[\/latex]. Let\u2019s factor [latex]125[\/latex] and find that number.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3\\;]{125}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>[latex]125[\/latex] ends in [latex]5[\/latex], so we know that \u00a0[latex]5[\/latex] is a factor. Expand [latex]125[\/latex] into [latex]5\\cdot25[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3\\;]{5\\cdot 25}[\/latex]<\/p>\n<p>Factor [latex]25[\/latex] into [latex]5\\cdot 5[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3\\;]{5\\cdot 5\\cdot 5}[\/latex]<\/p>\n<p>The factors are [latex]5\\cdot5\\cdot5[\/latex], or [latex]5^{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3\\;]{{{5}^{3}}}[\/latex]<\/p>\n<p>Answer:\u00a0 \u00a0 \u00a0 [latex]\\sqrt[3\\;]{125}=5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The prime factors of [latex]125[\/latex] are [latex]5\\cdot5\\cdot5[\/latex], which can be rewritten as [latex]5^{3}[\/latex]. The cube root of a cubed number is the number itself, so [latex]\\sqrt[3\\;]{{{5}^{3}}}=5[\/latex]. We have found the cube root, the three identical factors that when multiplied together give [latex]125[\/latex]. [latex]125[\/latex] is known as a <em><strong>perfect cube<\/strong><\/em> because its cube root is an integer.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Evaluate:<\/p>\n<ol>\n<li>[latex]\\sqrt[3\\;]{8}[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{-8}[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{-125}[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{64}[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{-1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm314\">Show Answer<\/span><\/p>\n<div id=\"qhjm314\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt[3\\;]{8}=2[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{-8}=-2[\/latex]<\/li>\n<li>[latex]-\\sqrt[3\\;]{-125}=5[\/latex]<\/li>\n<li>[latex]-\\sqrt[3\\;]{64}=-4[\/latex]<\/li>\n<li>[latex]\\sqrt[3\\;]{-1}=-1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">nth Roots<\/span><\/p>\n<p>We learned above that the cube root of a number is written with a small number\u00a0[latex]3[\/latex], which looks like [latex]\\sqrt[3\\;]{a}[\/latex]. This number placed just outside and above the radical symbol and is called the <strong><em>index<\/em>.<\/strong><\/p>\n<p>We can apply the same idea to any exponent and its corresponding root. 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 <em><strong>principal nth root<\/strong><\/em> 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\u00a0[latex]2[\/latex]. In the radical expression, [latex]n[\/latex] is called the <em><strong>index<\/strong><\/em> of the radical.<\/p>\n<div class=\"textbox shaded\">\n<h3>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<\/h3>\n<p>Evaluate each of the following:<\/p>\n<ol>\n<li>[latex]\\sqrt[5\\;]{-32}[\/latex]<\/li>\n<li>[latex]\\sqrt[4\\;]{81}[\/latex]<\/li>\n<li>[latex]\\sqrt[8\\;]{-1}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]\\sqrt[5\\;]{-32}[\/latex] Factoring [latex]32[\/latex] gives [latex]{\\left(-2\\right)}^{5}=-32\\text{ so }\\sqrt[5\\;]{-32}=-2[\/latex]<\/li>\n<li>[latex]\\sqrt[4\\;]{81}[\/latex]. Factoring can help. We know that [latex]9\\cdot9=81[\/latex] and we can further factor each\u00a0[latex]9[\/latex] into [latex]3\\cdot3[\/latex]. So, [latex]81={3^4}[\/latex] and therefore,\u00a0[latex]\\sqrt[4\\;]{81}=3[\/latex].<\/li>\n<li>[latex]\\sqrt[8\\;]{-1}[\/latex]. Since we have an\u00a0[latex]8[\/latex]th root with a negative number as the radicand, this root has no real number solutions. In other words, there is no real number that when raised to the power 8 results in a negative number.<\/li>\n<\/ol>\n<\/div>\n<p>The nth roots of a real number are categorized into two groups: <em><strong>odd roots<\/strong><\/em> when the index is an odd number, and <em><strong>even roots<\/strong><\/em> when the index is an even number. Square roots are even roots, since 2 is an even number. The square root of a negative real number is undefined in the set of real numbers. The same can be said for all even roots: the even root of a negative real number is undefined in the set of real numbers. This is because any real number raised to an even power is always positive. For example, [latex](-3)^4=81[\/latex]; [latex](-2)^6=64[\/latex]; [latex](-1)^126=1[\/latex].<\/p>\n<p>However, odd roots, like the cube root, exist for all real numbers. For example, [latex](-2)^5=-32[\/latex], so [latex]\\sqrt[5\\;]{-32}=-2[\/latex], and since [latex]4^3=64[\/latex], [latex]\\sqrt[3\\;]{64}=4[\/latex]. Odd roots take on the sign of the radicand.<\/p>\n<p>The steps to consider when simplifying a radical are outlined below.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplifying a radical<\/h3>\n<p>When working with exponents and radicals:<\/p>\n<ul>\n<li>If <i>n<\/i> is odd, [latex]\\sqrt[n\\;]{{{x}^{n}}}=x[\/latex].<\/li>\n<\/ul>\n<p style=\"padding-left: 60px;\">The root takes the sign of the radicand.<\/p>\n<ul>\n<li>If <i>n<\/i> is even, [latex]\\sqrt[n\\;]{{{x}^{n}}}=\\large | \\normalsize x \\large |[\/latex]. (The absolute value accounts for the fact that if [latex]x[\/latex] is negative and raised to an even power, that number will be positive, as will the <i>n<\/i>th principal root of that number.)<\/li>\n<\/ul>\n<p style=\"padding-left: 60px;\">The root is positive and the <em>n<\/em>th root of a negative radicand is undefined in the set of real numbers.<\/p>\n<\/div>\n<p>The video shows more examples of how to evaluate <em>n<\/em>th roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Perfect Nth Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vA2DkcUSRSk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We can find the odd root of a negative number, but we cannot find the even root of a negative number in the set of real numbers. This means we can evaluate the radicals [latex]\\sqrt[3\\;]{-81},\\ \\sqrt[5\\;]{-64}[\/latex], and [latex]\\sqrt[7\\;]{-2187}[\/latex] because they all have an odd numbered index, but we cannot evaluate the radicals\u00a0 [latex]\\sqrt[{}]{-100},\\ \\sqrt[4\\;]{-16}[\/latex], or [latex]\\sqrt[6\\;]{-2,500}[\/latex] because they all have an even numbered index.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm94221\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=94221&theme=oea&iframe_resize_id=ohm94221&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Rational Exponents<\/h2>\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. We can use rational exponents instead of a radical. A <em><strong>rational exponent<\/strong><\/em> is an exponent that is a fraction. For example, [latex]\\sqrt{4}[\/latex] can be written as [latex]{{4}^{\\small\\frac{1}{2}}}[\/latex].<\/p>\n<p>Radicals and rational 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>Use the example below to familiarize yourself with the different ways to write square roots.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Fill in the missing cells in the table.<\/p>\n<table style=\"width: 50%;\">\n<thead>\n<tr>\n<th>Exponent Form<\/th>\n<th>Root Form<\/th>\n<th>Root of a Square<\/th>\n<th>Simplified<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{{36}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\sqrt{81}[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>[latex]\\sqrt{{{12}^{2}}}[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q990781\">Show Solution<\/span><\/p>\n<div id=\"q990781\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 50%;\">\n<thead>\n<tr>\n<th>Exponent Form<\/th>\n<th>Root Form<\/th>\n<th>Root of a Square<\/th>\n<th>Simplified<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{{36}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{36}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{6}^{2}}}[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{{81}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{81}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{9}^{2}}}[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{{144}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>[latex]\\sqrt{144}[\/latex]<\/td>\n<td>[latex]\\sqrt{{{12}^{2}}}[\/latex]<\/td>\n<td>[latex]12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show another example of filling in a table to connect the different notation used for roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Write Basic Expression in Radical Form and Using Rational Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/eGJgmo2CpN4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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\\;]{x}[\/latex], and [latex]\\frac{1}{8}[\/latex] translates to the eighth root or [latex]\\sqrt[8\\;]{x}[\/latex].<\/p>\n<p>When converting from radical to rational exponent notation, the index of the root becomes the denominator of the exponent. If we start with a square root, we will have an exponent of [latex]\\frac{1}{2}[\/latex] on the radicand. On the other hand, if we start with an exponent of [latex]\\frac{1}{3}[\/latex] we will use a cube root. The following statement summarizes this idea.<\/p>\n<div class=\"textbox shaded\">\n<h3>Writing Fractional Exponents<\/h3>\n<p>Any radical in the form [latex]\\sqrt[n\\;]{a}[\/latex]\u00a0 can be written using a fractional exponent in the form [latex]a^{\\frac{1}{n}}[\/latex].<\/p>\n<\/div>\n<h2>Simplifying Nth Roots on a Calculator<\/h2>\n<p>In the same way we can find a square root on a calculator, we can find a higher root in the calculator to estimate what the root will be. For example, consider the [latex]\\sqrt[\\;5\\;]{250}[\/latex]. The calculator, depending on your type of calculator, has two different function keys that will take <em>n<\/em>th roots. The first is the key [latex]\\sqrt[y\\;]{x}[\/latex], the second is converting the root to a rational exponent and using the exponent key on the calculator.<\/p>\n<p>On the free online scientific calculator from Desmos, we find [latex]\\sqrt[7\\;]{128}[\/latex] by clicking on the nth root button, followed by the index 7, followed by the radicand, 150 after clicking under the radical in the display. The answer shows on the right side of the display and rounds to three decimal places as 2.046.<\/p>\n<p>We could get the same answer by writing\u00a0[latex]\\sqrt[7\\;]{128}[\/latex] as [latex]128^\\frac{1}{7}[\/latex] and clicking 128 then the exponent button [latex]a^b[\/latex] followed by the exponent [latex]1\\div 7[\/latex]. As we can see, we get the same answer.<\/p>\n<div id=\"attachment_1790\" style=\"width: 1034px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1790\" class=\"wp-image-1790 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/08\/23000604\/Desmos-nth-roots-1024x621.png\" alt=\"calculator image\" width=\"1024\" height=\"621\" \/><\/p>\n<p id=\"caption-attachment-1790\" class=\"wp-caption-text\">Free Desmos calculator available at desmos.com<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Use a calculator to evaluate the roots to 3 decimal places:<\/p>\n<p>1. [latex]\\sqrt[\\;6\\;]{768}[\/latex]<\/p>\n<p>2.\u00a0[latex]\\sqrt[\\;4\\;]{321}[\/latex]<\/p>\n<p>3.\u00a0[latex]\\sqrt[\\;3\\;]{34}[\/latex]<\/p>\n<p>4.\u00a0[latex]\\sqrt[\\;7\\;]{-7000}[\/latex]<\/p>\n<p>5.\u00a0[latex]\\sqrt[\\;5\\;]{-96}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm655\">Show Answer<\/span><\/p>\n<div id=\"qhjm655\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]3.026[\/latex]<\/li>\n<li>[latex]4.233[\/latex]<\/li>\n<li>[latex]3.240[\/latex]<\/li>\n<li>[latex]-3.542[\/latex]<\/li>\n<li>[latex]-2.491[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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-626\">\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>Screenshot: Desmos calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Caution. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify perfect n-th roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vA2DkcUSRSk\">https:\/\/youtu.be\/vA2DkcUSRSk<\/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>Write Basic Expression in Radical Form and Using Rational Exponents. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/eGJgmo2CpN4\">https:\/\/youtu.be\/eGJgmo2CpN4<\/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, Shared previously<\/div><ul class=\"citation-list\"><li>Edited and adapted: Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422605,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Screenshot: Desmos calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Caution\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Edited and adapted: Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Simplify perfect n-th roots\",\"author\":\"James Sousa 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