{"id":1743,"date":"2023-10-12T00:32:04","date_gmt":"2023-10-12T00:32:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-radicals-and-rational-exponents\/"},"modified":"2023-10-12T00:32:04","modified_gmt":"2023-10-12T00:32:04","slug":"introduction-radicals-and-rational-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-precalculus\/chapter\/introduction-radicals-and-rational-exponents\/","title":{"raw":"Radicals and Rational Exponents","rendered":"Radicals and Rational Exponents"},"content":{"raw":"\n\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li class=\"li2\"><span class=\"s1\">Evaluate and simplify square roots.<\/span><\/li>\n \t<li class=\"li2\"><span class=\"s1\">Rationalize a denominator that contains a square root.<\/span><\/li>\n \t<li class=\"li3\"><span class=\"s4\">Rewrite a radical expression using rational exponents.<\/span><\/li>\n<\/ul>\n<\/div>\nA hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203627\/CNX_CAT_Figure_01_03_001.jpg\" alt=\"A right triangle with a base of 5 feet, a height of 12 feet, and a hypotenuse labeled c\" width=\"487\" height=\"284\"> <b>Figure 1<\/b>[\/caption]\n<p style=\"text-align: center\">[latex]\\begin{align} {a}^{2}+{b}^{2}&amp; = {c}^{2} \\\\ {5}^{2}+{12}^{2}&amp; = {c}^{2} \\\\ 169&amp; = {c}^{2} \\end{align}[\/latex]<\/p>\nNow we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other word we need to find a square root. In this section we will investigate methods of finding solutions to problems such as this one.\n<h2>Evaluate and Simplify Square Roots<\/h2>\nWhen the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[\/latex], the square root of [latex]16[\/latex] is [latex]4[\/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.\n\nIn general terms, if [latex]a[\/latex] is a positive real number, then the square root of [latex]a[\/latex] is a number that, when multiplied by itself, gives [latex]a[\/latex]. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The <strong>principal square root<\/strong> is the nonnegative number that when multiplied by itself equals [latex]a[\/latex]. The square root obtained using a calculator is the principal square root.\n\nThe principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a <strong>radical<\/strong>, the term under the symbol is called the <strong>radicand<\/strong>, and the entire expression is called a <strong>radical expression<\/strong>.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\">\n<div class=\"textbox\">\n<h3>A General Note: Principal Square Root<\/h3>\nThe <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a <strong>radical expression<\/strong>, with a symbol called a <strong>radical<\/strong> over the term called the <strong>radicand<\/strong>: [latex]\\sqrt{a}[\/latex].\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3><strong>Does<\/strong> [latex]\\sqrt{25}=\\pm 5[\/latex]?<\/h3>\n<em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Square Roots<\/h3>\nEvaluate each expression.\n<ol>\n \t<li>[latex]\\sqrt{100}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"849035\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"849035\"]\n<ol>\n \t<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\n \t<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\n \t<li>[latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n \t<li>[latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/h3>\n<em>No.<\/em> [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. <em>This is not equivalent to<\/em> [latex]\\sqrt{25+144}=13[\/latex]. <em>The order of operations requires us to add the terms in the radicand before finding the square root.<\/em>\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nEvaluate each expression.\n<ol>\n \t<li>[latex]\\sqrt{225}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{\\sqrt{81}}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{25 - 9}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"98241\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"98241\"]\n<ol>\n \t<li>[latex]15[\/latex]<\/li>\n \t<li>[latex]3[\/latex]<\/li>\n \t<li>[latex]4[\/latex]<\/li>\n \t<li>[latex]17[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14119&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109776&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<h3>Use the Product Rule to Simplify Square Roots<\/h3>\nTo simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the <em>product rule for simplifying square roots,<\/em> which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\\sqrt{15}[\/latex] as [latex]\\sqrt{3}\\cdot \\sqrt{5}[\/latex]. We can also use the product rule to express the product of multiple radical expressions as a single radical expression.\n<div class=\"textbox\">\n<h3>A General Note: The Product Rule for Simplifying Square Roots<\/h3>\nIf [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the product [latex]ab[\/latex] is equal to the product of the square roots of [latex]a[\/latex] and [latex]b[\/latex].\n<div style=\"text-align: center\">[latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a square root radical expression, use the product rule to simplify it.<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Factor any perfect squares from the radicand.<\/li>\n \t<li>Write the radical expression as a product of radical expressions.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule to Simplify Square Roots<\/h3>\nSimplify the radical expression.\n<ol>\n \t<li>[latex]\\sqrt{300}[\/latex]<\/li>\n \t<li>[latex]\\sqrt{162{a}^{5}{b}^{4}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"483887\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"483887\"]\n\n1.\n[latex]\\begin{align}&amp;\\sqrt{100\\cdot 3} &amp;&amp; \\text{Factor perfect square from radicand}. \\\\ &amp;\\sqrt{100}\\cdot \\sqrt{3} &amp;&amp; \\text{Write radical expression as product of radical expressions}. \\\\ &amp;10\\sqrt{3} &amp;&amp; \\text{Simplify}. \\\\ \\text{ }\\end{align}[\/latex]\n\n2.\n[latex]\\begin{align}&amp;\\sqrt{81{a}^{4}{b}^{4}\\cdot 2a} &amp;&amp; \\text{Factor perfect square from radicand}. \\\\ &amp;\\sqrt{81{a}^{4}{b}^{4}}\\cdot \\sqrt{2a} &amp;&amp; \\text{Write radical expression as product of radical expressions}. \\\\ &amp;9{a}^{2}{b}^{2}\\sqrt{2a} &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify [latex]\\sqrt{50{x}^{2}{y}^{3}z}[\/latex].\n\n[reveal-answer q=\"86548\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"86548\"]\n\n[latex]5|x||y|\\sqrt{2yz}[\/latex]. Notice the absolute value signs around <em>x<\/em> and <em>y<\/em>? That\u2019s because their value must be positive!\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110285&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Express the product of multiple radical expressions as a single radical expression.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule to Simplify the Product of Multiple Square Roots<\/h3>\nSimplify the radical expression.\n<p style=\"text-align: center\">[latex]\\sqrt{12}\\cdot \\sqrt{3}[\/latex]<\/p>\n[reveal-answer q=\"134287\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"134287\"]\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;\\sqrt{12\\cdot 3} &amp;&amp; \\text{Express the product as a single radical expression}. \\\\ &amp;\\sqrt{36} &amp;&amp; \\text{Simplify}. \\\\ &amp;6 \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify [latex]\\sqrt{50x}\\cdot \\sqrt{2x}[\/latex] assuming [latex]x&gt;0[\/latex].\n\n[reveal-answer q=\"481919\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"481919\"]\n\n[latex]10x[\/latex]\nBecause [latex]x&gt;0[\/latex], we do not need an absolute values.\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110272&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Using the Quotient Rule to Simplify Square Roots<\/h2>\nJust as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the <em>quotient rule for simplifying square roots.<\/em> It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\\sqrt{\\dfrac{5}{2}}[\/latex] as [latex]\\dfrac{\\sqrt{5}}{\\sqrt{2}}[\/latex].\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule for Simplifying Square Roots<\/h3>\nThe square root of the quotient [latex]\\dfrac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex], where [latex]b\\ne 0[\/latex].\n<div style=\"text-align: center\">[latex]\\sqrt{\\dfrac{a}{b}}=\\dfrac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a radical expression, use the quotient rule to simplify it.<strong>\n<\/strong><\/h3>\n<ol>\n \t<li>Write the radical expression as the quotient of two radical expressions.<\/li>\n \t<li>Simplify the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule to Simplify Square Roots<\/h3>\nSimplify the radical expression.\n<p style=\"text-align: center\">[latex]\\sqrt{\\dfrac{5}{36}}[\/latex]<\/p>\n[reveal-answer q=\"317945\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"317945\"]\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;\\frac{\\sqrt{5}}{\\sqrt{36}} &amp;&amp; \\text{Write as quotient of two radical expressions}. \\\\ &amp;\\frac{\\sqrt{5}}{6} &amp;&amp; \\text{Simplify denominator}. \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify [latex]\\sqrt{\\dfrac{2{x}^{2}}{9{y}^{4}}}[\/latex].\n\n[reveal-answer q=\"671876\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"671876\"]\n\n[latex]\\dfrac{x\\sqrt{2}}{3{y}^{2}}[\/latex]. We do not need the absolute value signs for [latex]{y}^{2}[\/latex] because that term will always be nonnegative.\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110287&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule to Simplify an Expression with Two Square Roots<\/h3>\nSimplify the radical expression.\n<p style=\"text-align: center\">[latex]\\dfrac{\\sqrt{234{x}^{11}y}}{\\sqrt{26{x}^{7}y}}[\/latex]<\/p>\n[reveal-answer q=\"520119\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"520119\"]\n<p style=\"text-align: center\">[latex]\\begin{align}&amp;\\sqrt{\\frac{234{x}^{11}y}{26{x}^{7}y}} &amp;&amp; \\text{Combine numerator and denominator into one radical expression}. \\\\ &amp;\\sqrt{9{x}^{4}} &amp;&amp; \\text{Simplify fraction}. \\\\ &amp;3{x}^{2} &amp;&amp; \\text{Simplify square root}. \\end{align}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify [latex]\\dfrac{\\sqrt{9{a}^{5}{b}^{14}}}{\\sqrt{3{a}^{4}{b}^{5}}}[\/latex].\n\n[reveal-answer q=\"157179\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"157179\"]\n\n[latex]{b}^{4}\\sqrt{3ab}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110387&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\nIn the following video you will see more examples of how to simplify radical expressions with variables.\nhttps:\/\/youtu.be\/q7LqsKPoAKo\n<h2>Operations on Square Roots<\/h2>\nWe can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}[\/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2}[\/latex], so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}[\/latex].\n<div class=\"textbox\">\n<h3>How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\n<ol>\n \t<li>Simplify each radical expression.<\/li>\n \t<li>Add or subtract expressions with equal radicands.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Square Roots<\/h3>\nAdd [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex].\n\n[reveal-answer q=\"742464\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"742464\"]\n\nWe can rewrite [latex]5\\sqrt{12}[\/latex] as [latex]5\\sqrt{4\\cdot 3}[\/latex]. According the product rule, this becomes [latex]5\\sqrt{4}\\sqrt{3}[\/latex]. The square root of [latex]\\sqrt{4}[\/latex] is 2, so the expression becomes [latex]5\\left(2\\right)\\sqrt{3}[\/latex], which is [latex]10\\sqrt{3}[\/latex]. Now we can the terms have the same radicand so we can add.\n<p style=\"text-align: center\">[latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nAdd [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].\n\n[reveal-answer q=\"21382\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"21382\"]\n\n[latex]13\\sqrt{5}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2049&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\nWatch this video to see more examples of adding roots.\nhttps:\/\/youtu.be\/S3fGUeALy7E\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Square Roots<\/h3>\nSubtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].\n\n[reveal-answer q=\"902648\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"902648\"]\n\nRewrite each term so they have equal radicands.\n<div style=\"text-align: center\">[latex]\\begin{align} 20\\sqrt{72{a}^{3}{b}^{4}c}&amp; = 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ &amp; = 20\\left(3\\right)\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ &amp; = 120a{b}^{2}\\sqrt{2ac}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: center\">[latex]\\begin{align} 14\\sqrt{8{a}^{3}{b}^{4}c}&amp; = 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ &amp; = 14\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ &amp; = 28a{b}^{2}\\sqrt{2ac} \\end{align}[\/latex]<\/div>\nNow the terms have the same radicand so we can subtract.\n<div>[latex]120a{b}^{2}\\sqrt{2ac}-28a{b}^{2}\\sqrt{2ac}=92a{b}^{2}\\sqrt{2ac} \\\\ [\/latex]<\/div>\n<div>Note that we do not need an absolute value around the <em>a<\/em> because the [latex]a^3[\/latex] under the radical means that&nbsp;<em>a<\/em> can't be negative.<\/div>\n<div>[\/hidden-answer]<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSubtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].\n\n[reveal-answer q=\"236912\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"236912\"]\n\n[latex]0[\/latex]\n\n[\/hidden-answer]\n\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110419&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\nin the next video we show more examples of how to subtract radicals.\nhttps:\/\/youtu.be\/77TR9HsPZ6M\n<h2>Rationalize Denominators<\/h2>\nWhen an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em>rationalizing the denominator<\/em>.\n\nWe know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.\n\nFor a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\\sqrt{c}[\/latex], multiply by [latex]\\dfrac{\\sqrt{c}}{\\sqrt{c}}[\/latex].\n\nFor a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\\sqrt{c}[\/latex], then the conjugate is [latex]a-b\\sqrt{c}[\/latex].\n<div class=\"textbox\">\n<h3>How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/h3>\n<ol>\n \t<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing a Single Term<\/h3>\nWrite [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.\n\n[reveal-answer q=\"982148\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"982148\"]\n\nThe radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.\n<div style=\"text-align: center\">[latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &amp;= \\frac{2\\sqrt{30}}{30} \\\\ &amp;= \\frac{\\sqrt{30}}{15}\\end{align}[\/latex]<\/div>\n<div>[\/hidden-answer]<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.\n\n[reveal-answer q=\"497322\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"497322\"]\n\n[latex]6\\sqrt{6}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2765&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/h3>\n<ol>\n \t<li>Find the conjugate of the denominator.<\/li>\n \t<li>Multiply the numerator and denominator by the conjugate.<\/li>\n \t<li>Use the distributive property.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing Two Terms<\/h3>\nWrite [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex] in simplest form.\n\n[reveal-answer q=\"726340\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"726340\"]\n\nBegin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}[\/latex]. Then multiply the fraction by [latex]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].\n<div style=\"text-align: center\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &amp;= \\frac{4 - 4\\sqrt{5}}{-4} &amp;&amp; \\text{Use the distributive property}. \\\\ &amp;=\\sqrt{5}-1 &amp;&amp; \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<div>[\/hidden-answer]<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite [latex]\\frac{7}{2+\\sqrt{3}}[\/latex] in simplest form.\n\n[reveal-answer q=\"132932\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"132932\"]\n\n[latex]14 - 7\\sqrt{3}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>N<sup>th<\/sup> Roots and Rational Exponents<\/h2>\n<h3>Using Rational Roots<\/h3>\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.\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.\n\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].\n\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.\n<div class=\"textbox\">\n<h3>A General Note: Principal <em>n<\/em>th Root<\/h3>\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].\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying <em>n<\/em>th Roots<\/h3>\nSimplify each of the following:\n<ol>\n \t<li>[latex]\\sqrt[5]{-32}[\/latex]<\/li>\n \t<li>[latex]\\sqrt[4]{4}\\cdot \\sqrt[4]{1,024}[\/latex]<\/li>\n \t<li>[latex]-\\sqrt[3]{\\dfrac{8{x}^{6}}{125}}[\/latex]<\/li>\n \t<li>[latex]8\\sqrt[4]{3}-\\sqrt[4]{48}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"149528\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"149528\"]\n<ol>\n \t<li>[latex]\\sqrt[5]{-32}=-2[\/latex] because [latex]{\\left(-2\\right)}^{5}=-32 \\\\ \\text{ }[\/latex]<\/li>\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>\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>\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>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify.\n<ol>\n \t<li>[latex]\\sqrt[3]{-216}[\/latex]<\/li>\n \t<li>[latex]\\dfrac{3\\sqrt[4]{80}}{\\sqrt[4]{5}}[\/latex]<\/li>\n \t<li>[latex]6\\sqrt[3]{9,000}+7\\sqrt[3]{576}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"15987\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"15987\"]\n<ol>\n \t<li>[latex]-6[\/latex]<\/li>\n \t<li>[latex]6[\/latex]<\/li>\n \t<li>[latex]88\\sqrt[3]{9}[\/latex]<\/li>\n<\/ol>\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2564&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2565&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2567&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe>\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2592&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h3>Using Rational Exponents<\/h3>\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.\n<div style=\"text-align: center\">[latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\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.\n<div style=\"text-align: center\">[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\nAll of the properties of exponents that we learned for integer exponents also hold for rational exponents.\n<div class=\"textbox\">\n<h3>Rational Exponents<\/h3>\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\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 \t<li>Determine the power by looking at the numerator of the exponent.<\/li>\n \t<li>Determine the root by looking at the denominator of the exponent.<\/li>\n \t<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>\nWrite [latex]{343}^{\\frac{2}{3}}[\/latex] as a radical. Simplify.\n\n[reveal-answer q=\"878113\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"878113\"]\nThe 2 tells us the power and the 3 tells us the root.\n<p style=\"text-align: center\">[latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}=\\sqrt[3]{{343}^{2}}[\/latex]<\/p>\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.\n<p style=\"text-align: center\">[latex]{343}^{\\frac{2}{3}}={\\left(\\sqrt[3]{343}\\right)}^{2}={7}^{2}=49[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite [latex]{9}^{\\frac{5}{2}}[\/latex] as a radical. Simplify.\n\n[reveal-answer q=\"937831\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"937831\"]\n\n[latex]{\\left(\\sqrt{9}\\right)}^{5}={3}^{5}=243[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3415&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing Radicals as Rational Exponents<\/h3>\nWrite [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.\n\n[reveal-answer q=\"183909\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"183909\"]\n\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].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nWrite [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.\n\n[reveal-answer q=\"522860\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"522860\"]\n\n[latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\nWatch this video to see more examples of how to write a radical with a fractional exponent.\n\nhttps:\/\/youtu.be\/L5Z_3RrrVjA\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Rational Exponents<\/h3>\nSimplify:\n<ol>\n \t<li>[latex]5\\left(2{x}^{\\frac{3}{4}}\\right)\\left(3{x}^{\\frac{1}{5}}\\right)[\/latex]<\/li>\n \t<li>[latex]{\\left(\\dfrac{16}{9}\\right)}^{-\\frac{1}{2}}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"803060\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"803060\"]\n\n1.\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]\n\n2.\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]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nSimplify [latex]{\\left(8x\\right)}^{\\frac{1}{3}}\\left(14{x}^{\\frac{6}{5}}\\right)[\/latex].\n\n[reveal-answer q=\"95703\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"95703\"]\n\n[latex]28{x}^{\\frac{23}{15}}[\/latex]\n\n[\/hidden-answer]\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=59783&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe>\n\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>The principal square root of a number [latex]a[\/latex] is the nonnegative number that when multiplied by itself equals [latex]a[\/latex].<\/li>\n \t<li>If [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the product [latex]ab[\/latex] is equal to the product of the square roots of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n \t<li>If [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the quotient [latex]\\frac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n \t<li>We can add and subtract radical expressions if they have the same radicand and the same index.<\/li>\n \t<li>Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.<\/li>\n \t<li>The principal <em>n<\/em>th root 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]. These roots have the same properties as square roots.<\/li>\n \t<li>Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals.<\/li>\n \t<li>The properties of exponents apply to rational exponents.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<strong>index<\/strong> the number above the radical sign indicating the <em>n<\/em>th root\n\n<strong>principal <em>n<\/em>th root<\/strong> the number with the same sign as [latex]a[\/latex] that when raised to the <em>n<\/em>th power equals [latex]a[\/latex]\n\n<strong>principal square root<\/strong> the nonnegative square root of a number [latex]a[\/latex] that, when multiplied by itself, equals [latex]a[\/latex]\n\n<strong>radical<\/strong> the symbol used to indicate a root\n\n<strong>radical expression<\/strong> an expression containing a radical symbol\n\n<strong>radicand<\/strong> the number under the radical symbol\n\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li class=\"li2\"><span class=\"s1\">Evaluate and simplify square roots.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Rationalize a denominator that contains a square root.<\/span><\/li>\n<li class=\"li3\"><span class=\"s4\">Rewrite a radical expression using rational exponents.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203627\/CNX_CAT_Figure_01_03_001.jpg\" alt=\"A right triangle with a base of 5 feet, a height of 12 feet, and a hypotenuse labeled c\" width=\"487\" height=\"284\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p style=\"text-align: center\">[latex]\\begin{align} {a}^{2}+{b}^{2}& = {c}^{2} \\\\ {5}^{2}+{12}^{2}& = {c}^{2} \\\\ 169& = {c}^{2} \\end{align}[\/latex]<\/p>\n<p>Now we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other word we need to find a square root. In this section we will investigate methods of finding solutions to problems such as this one.<\/p>\n<h2>Evaluate and Simplify Square Roots<\/h2>\n<p>When the square root of a number is squared, the result is the original number. Since [latex]{4}^{2}=16[\/latex], the square root of [latex]16[\/latex] is [latex]4[\/latex]. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.<\/p>\n<p>In general terms, if [latex]a[\/latex] is a positive real number, then the square root of [latex]a[\/latex] is a number that, when multiplied by itself, gives [latex]a[\/latex]. The square root could be positive or negative because multiplying two negative numbers gives a positive number. The <strong>principal square root<\/strong> is the nonnegative number that when multiplied by itself equals [latex]a[\/latex]. The square root obtained using a calculator is the principal square root.<\/p>\n<p>The principal square root of [latex]a[\/latex] is written as [latex]\\sqrt{a}[\/latex]. The symbol is called a <strong>radical<\/strong>, the term under the symbol is called the <strong>radicand<\/strong>, and the entire expression is called a <strong>radical expression<\/strong>.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24203630\/CNX_CAT_Figure_01_03_002.jpg\" alt=\"The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: Principal Square Root<\/h3>\n<p>The <strong>principal square root<\/strong> of [latex]a[\/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[\/latex]. It is written as a <strong>radical expression<\/strong>, with a symbol called a <strong>radical<\/strong> over the term called the <strong>radicand<\/strong>: [latex]\\sqrt{a}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3><strong>Does<\/strong> [latex]\\sqrt{25}=\\pm 5[\/latex]?<\/h3>\n<p><em>No. Although both<\/em> [latex]{5}^{2}[\/latex] <em>and<\/em> [latex]{\\left(-5\\right)}^{2}[\/latex] <em>are<\/em> [latex]25[\/latex], <em>the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is<\/em> [latex]\\sqrt{25}=5[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating Square Roots<\/h3>\n<p>Evaluate each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{100}[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q849035\">Show Solution<\/span><\/p>\n<div id=\"q849035\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\sqrt{100}=10[\/latex] because [latex]{10}^{2}=100[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{16}}=\\sqrt{4}=2[\/latex] because [latex]{4}^{2}=16[\/latex] and [latex]{2}^{2}=4[\/latex]<\/li>\n<li>[latex]\\sqrt{25+144}=\\sqrt{169}=13[\/latex] because [latex]{13}^{2}=169[\/latex]<\/li>\n<li>[latex]\\sqrt{49}-\\sqrt{81}=7 - 9=-2[\/latex] because [latex]{7}^{2}=49[\/latex] and [latex]{9}^{2}=81[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>For [latex]\\sqrt{25+144}[\/latex], can we find the square roots before adding?<\/h3>\n<p><em>No.<\/em> [latex]\\sqrt{25}+\\sqrt{144}=5+12=17[\/latex]. <em>This is not equivalent to<\/em> [latex]\\sqrt{25+144}=13[\/latex]. <em>The order of operations requires us to add the terms in the radicand before finding the square root.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Evaluate each expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{225}[\/latex]<\/li>\n<li>[latex]\\sqrt{\\sqrt{81}}[\/latex]<\/li>\n<li>[latex]\\sqrt{25 - 9}[\/latex]<\/li>\n<li>[latex]\\sqrt{36}+\\sqrt{121}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q98241\">Show Solution<\/span><\/p>\n<div id=\"q98241\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]15[\/latex]<\/li>\n<li>[latex]3[\/latex]<\/li>\n<li>[latex]4[\/latex]<\/li>\n<li>[latex]17[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=14119&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=109776&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Use the Product Rule to Simplify Square Roots<\/h3>\n<p>To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the <em>product rule for simplifying square roots,<\/em> which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\\sqrt{15}[\/latex] as [latex]\\sqrt{3}\\cdot \\sqrt{5}[\/latex]. We can also use the product rule to express the product of multiple radical expressions as a single radical expression.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Product Rule for Simplifying Square Roots<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the product [latex]ab[\/latex] is equal to the product of the square roots of [latex]a[\/latex] and [latex]b[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]\\sqrt{ab}=\\sqrt{a}\\cdot \\sqrt{b}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a square root radical expression, use the product rule to simplify it.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Factor any perfect squares from the radicand.<\/li>\n<li>Write the radical expression as a product of radical expressions.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule to Simplify Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<ol>\n<li>[latex]\\sqrt{300}[\/latex]<\/li>\n<li>[latex]\\sqrt{162{a}^{5}{b}^{4}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q483887\">Show Solution<\/span><\/p>\n<div id=\"q483887\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<br \/>\n[latex]\\begin{align}&\\sqrt{100\\cdot 3} && \\text{Factor perfect square from radicand}. \\\\ &\\sqrt{100}\\cdot \\sqrt{3} && \\text{Write radical expression as product of radical expressions}. \\\\ &10\\sqrt{3} && \\text{Simplify}. \\\\ \\text{ }\\end{align}[\/latex]<\/p>\n<p>2.<br \/>\n[latex]\\begin{align}&\\sqrt{81{a}^{4}{b}^{4}\\cdot 2a} && \\text{Factor perfect square from radicand}. \\\\ &\\sqrt{81{a}^{4}{b}^{4}}\\cdot \\sqrt{2a} && \\text{Write radical expression as product of radical expressions}. \\\\ &9{a}^{2}{b}^{2}\\sqrt{2a} && \\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]\\sqrt{50{x}^{2}{y}^{3}z}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q86548\">Show Solution<\/span><\/p>\n<div id=\"q86548\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]5|x||y|\\sqrt{2yz}[\/latex]. Notice the absolute value signs around <em>x<\/em> and <em>y<\/em>? That\u2019s because their value must be positive!<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110285&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Express the product of multiple radical expressions as a single radical expression.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Product Rule to Simplify the Product of Multiple Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{12}\\cdot \\sqrt{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q134287\">Show Solution<\/span><\/p>\n<div id=\"q134287\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{align}&\\sqrt{12\\cdot 3} && \\text{Express the product as a single radical expression}. \\\\ &\\sqrt{36} && \\text{Simplify}. \\\\ &6 \\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]\\sqrt{50x}\\cdot \\sqrt{2x}[\/latex] assuming [latex]x>0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q481919\">Show Solution<\/span><\/p>\n<div id=\"q481919\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]10x[\/latex]<br \/>\nBecause [latex]x>0[\/latex], we do not need an absolute values.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110272&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Using the Quotient Rule to Simplify Square Roots<\/h2>\n<p>Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the <em>quotient rule for simplifying square roots.<\/em> It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\\sqrt{\\dfrac{5}{2}}[\/latex] as [latex]\\dfrac{\\sqrt{5}}{\\sqrt{2}}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Quotient Rule for Simplifying Square Roots<\/h3>\n<p>The square root of the quotient [latex]\\dfrac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex], where [latex]b\\ne 0[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]\\sqrt{\\dfrac{a}{b}}=\\dfrac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a radical expression, use the quotient rule to simplify it.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Write the radical expression as the quotient of two radical expressions.<\/li>\n<li>Simplify the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule to Simplify Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{\\dfrac{5}{36}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q317945\">Show Solution<\/span><\/p>\n<div id=\"q317945\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{align}&\\frac{\\sqrt{5}}{\\sqrt{36}} && \\text{Write as quotient of two radical expressions}. \\\\ &\\frac{\\sqrt{5}}{6} && \\text{Simplify denominator}. \\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]\\sqrt{\\dfrac{2{x}^{2}}{9{y}^{4}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q671876\">Show Solution<\/span><\/p>\n<div id=\"q671876\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{x\\sqrt{2}}{3{y}^{2}}[\/latex]. We do not need the absolute value signs for [latex]{y}^{2}[\/latex] because that term will always be nonnegative.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110287&amp;theme=oea&amp;iframe_resize_id=mom20\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Quotient Rule to Simplify an Expression with Two Square Roots<\/h3>\n<p>Simplify the radical expression.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{\\sqrt{234{x}^{11}y}}{\\sqrt{26{x}^{7}y}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q520119\">Show Solution<\/span><\/p>\n<div id=\"q520119\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]\\begin{align}&\\sqrt{\\frac{234{x}^{11}y}{26{x}^{7}y}} && \\text{Combine numerator and denominator into one radical expression}. \\\\ &\\sqrt{9{x}^{4}} && \\text{Simplify fraction}. \\\\ &3{x}^{2} && \\text{Simplify square root}. \\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]\\dfrac{\\sqrt{9{a}^{5}{b}^{14}}}{\\sqrt{3{a}^{4}{b}^{5}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q157179\">Show Solution<\/span><\/p>\n<div id=\"q157179\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{b}^{4}\\sqrt{3ab}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110387&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>In the following video you will see more examples of how to simplify radical expressions with variables.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Square Roots with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q7LqsKPoAKo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Operations on Square Roots<\/h2>\n<p>We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\\sqrt{2}[\/latex] and [latex]3\\sqrt{2}[\/latex] is [latex]4\\sqrt{2}[\/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\\sqrt{18}[\/latex] can be written with a [latex]2[\/latex] in the radicand, as [latex]3\\sqrt{2}[\/latex], so [latex]\\sqrt{2}+\\sqrt{18}=\\sqrt{2}+3\\sqrt{2}=4\\sqrt{2}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a radical expression requiring addition or subtraction of square roots, solve.<\/h3>\n<ol>\n<li>Simplify each radical expression.<\/li>\n<li>Add or subtract expressions with equal radicands.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Square Roots<\/h3>\n<p>Add [latex]5\\sqrt{12}+2\\sqrt{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q742464\">Show Solution<\/span><\/p>\n<div id=\"q742464\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can rewrite [latex]5\\sqrt{12}[\/latex] as [latex]5\\sqrt{4\\cdot 3}[\/latex]. According the product rule, this becomes [latex]5\\sqrt{4}\\sqrt{3}[\/latex]. The square root of [latex]\\sqrt{4}[\/latex] is 2, so the expression becomes [latex]5\\left(2\\right)\\sqrt{3}[\/latex], which is [latex]10\\sqrt{3}[\/latex]. Now we can the terms have the same radicand so we can add.<\/p>\n<p style=\"text-align: center\">[latex]10\\sqrt{3}+2\\sqrt{3}=12\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Add [latex]\\sqrt{5}+6\\sqrt{20}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q21382\">Show Solution<\/span><\/p>\n<div id=\"q21382\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]13\\sqrt{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2049&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of adding roots.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Adding Radicals That Requires Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/S3fGUeALy7E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Subtracting Square Roots<\/h3>\n<p>Subtract [latex]20\\sqrt{72{a}^{3}{b}^{4}c}-14\\sqrt{8{a}^{3}{b}^{4}c}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q902648\">Show Solution<\/span><\/p>\n<div id=\"q902648\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite each term so they have equal radicands.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align} 20\\sqrt{72{a}^{3}{b}^{4}c}& = 20\\sqrt{9}\\sqrt{4}\\sqrt{2}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ & = 20\\left(3\\right)\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ & = 120a{b}^{2}\\sqrt{2ac}\\\\ \\text{ } \\end{align}[\/latex]<\/div>\n<div><\/div>\n<div style=\"text-align: center\">[latex]\\begin{align} 14\\sqrt{8{a}^{3}{b}^{4}c}& = 14\\sqrt{2}\\sqrt{4}\\sqrt{a}\\sqrt{{a}^{2}}\\sqrt{{\\left({b}^{2}\\right)}^{2}}\\sqrt{c} \\\\ & = 14\\left(2\\right)a{b}^{2}\\sqrt{2ac} \\\\ & = 28a{b}^{2}\\sqrt{2ac} \\end{align}[\/latex]<\/div>\n<p>Now the terms have the same radicand so we can subtract.<\/p>\n<div>[latex]120a{b}^{2}\\sqrt{2ac}-28a{b}^{2}\\sqrt{2ac}=92a{b}^{2}\\sqrt{2ac} \\\\[\/latex]<\/div>\n<div>Note that we do not need an absolute value around the <em>a<\/em> because the [latex]a^3[\/latex] under the radical means that&nbsp;<em>a<\/em> can&#8217;t be negative.<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract [latex]3\\sqrt{80x}-4\\sqrt{45x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236912\">Show Solution<\/span><\/p>\n<div id=\"q236912\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110419&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>in the next video we show more examples of how to subtract radicals.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Subtracting Radicals (Basic With No Simplifying)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/77TR9HsPZ6M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Rationalize Denominators<\/h2>\n<p>When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called <em>rationalizing the denominator<\/em>.<\/p>\n<p>We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.<\/p>\n<p>For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is [latex]b\\sqrt{c}[\/latex], multiply by [latex]\\dfrac{\\sqrt{c}}{\\sqrt{c}}[\/latex].<\/p>\n<p>For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is [latex]a+b\\sqrt{c}[\/latex], then the conjugate is [latex]a-b\\sqrt{c}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a single square root radical term in the denominator, rationalize the denominator.<\/h3>\n<ol>\n<li>Multiply the numerator and denominator by the radical in the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing a Single Term<\/h3>\n<p>Write [latex]\\dfrac{2\\sqrt{3}}{3\\sqrt{10}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982148\">Show Solution<\/span><\/p>\n<div id=\"q982148\" class=\"hidden-answer\" style=\"display: none\">\n<p>The radical in the denominator is [latex]\\sqrt{10}[\/latex]. So multiply the fraction by [latex]\\dfrac{\\sqrt{10}}{\\sqrt{10}}[\/latex]. Then simplify.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}\\frac{2\\sqrt{3}}{3\\sqrt{10}}\\cdot \\frac{\\sqrt{10}}{\\sqrt{10}} &= \\frac{2\\sqrt{30}}{30} \\\\ &= \\frac{\\sqrt{30}}{15}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\dfrac{12\\sqrt{3}}{\\sqrt{2}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497322\">Show Solution<\/span><\/p>\n<div id=\"q497322\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2765&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a radical term and a constant in the denominator, rationalize the denominator.<\/h3>\n<ol>\n<li>Find the conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator by the conjugate.<\/li>\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Rationalizing a Denominator Containing Two Terms<\/h3>\n<p>Write [latex]\\dfrac{4}{1+\\sqrt{5}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q726340\">Show Solution<\/span><\/p>\n<div id=\"q726340\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of [latex]1+\\sqrt{5}[\/latex] is [latex]1-\\sqrt{5}[\/latex]. Then multiply the fraction by [latex]\\dfrac{1-\\sqrt{5}}{1-\\sqrt{5}}[\/latex].<\/p>\n<div style=\"text-align: center\">[latex]\\begin{align}\\frac{4}{1+\\sqrt{5}}\\cdot \\frac{1-\\sqrt{5}}{1-\\sqrt{5}} &= \\frac{4 - 4\\sqrt{5}}{-4} && \\text{Use the distributive property}. \\\\ &=\\sqrt{5}-1 && \\text{Simplify}. \\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\frac{7}{2+\\sqrt{3}}[\/latex] in simplest form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q132932\">Show Solution<\/span><\/p>\n<div id=\"q132932\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]14 - 7\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3441&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>N<sup>th<\/sup> Roots and Rational Exponents<\/h2>\n<h3>Using Rational Roots<\/h3>\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=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2564&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2565&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2567&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"350\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2592&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h3>Using Rational Exponents<\/h3>\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=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3415&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/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-4\" 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><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=59783&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The principal square root of a number [latex]a[\/latex] is the nonnegative number that when multiplied by itself equals [latex]a[\/latex].<\/li>\n<li>If [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the product [latex]ab[\/latex] is equal to the product of the square roots of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<li>If [latex]a[\/latex] and [latex]b[\/latex] are nonnegative, the square root of the quotient [latex]\\frac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex]<\/li>\n<li>We can add and subtract radical expressions if they have the same radicand and the same index.<\/li>\n<li>Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.<\/li>\n<li>The principal <em>n<\/em>th root 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]. These roots have the same properties as square roots.<\/li>\n<li>Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals.<\/li>\n<li>The properties of exponents apply to rational exponents.<\/li>\n<\/ul>\n<div>\n<h2>Glossary<\/h2>\n<p><strong>index<\/strong> the number above the radical sign indicating the <em>n<\/em>th root<\/p>\n<p><strong>principal <em>n<\/em>th root<\/strong> 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><strong>principal square root<\/strong> the nonnegative square root of a number [latex]a[\/latex] that, when multiplied by itself, equals [latex]a[\/latex]<\/p>\n<p><strong>radical<\/strong> the symbol used to indicate a root<\/p>\n<p><strong>radical expression<\/strong> an expression containing a radical symbol<\/p>\n<p><strong>radicand<\/strong> the number under the radical symbol<\/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-1743\">\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>Simplify Square Roots With Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q7LqsKPoAKo\">https:\/\/youtu.be\/q7LqsKPoAKo<\/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 14119. <strong>Authored by<\/strong>: James Sousa. <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 109776, 110285, 110272, 110287, 110387. <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>Adding Radicals Requiring Simplification. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/S3fGUeALy7E\">https:\/\/youtu.be\/S3fGUeALy7E<\/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>Subtracting Radicals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/77TR9HsPZ6M\">https:\/\/youtu.be\/77TR9HsPZ6M<\/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 2049. <strong>Authored by<\/strong>: Lawrence Morales. <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 110419. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 2765. <strong>Authored by<\/strong>: Bryan Johns. <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 3441, 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>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 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 <\/section>","protected":false},"author":708740,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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