{"id":642,"date":"2024-05-30T18:36:25","date_gmt":"2024-05-30T18:36:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/?post_type=chapter&#038;p=642"},"modified":"2024-07-29T15:21:45","modified_gmt":"2024-07-29T15:21:45","slug":"5-4-dividing-radical-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/5-4-dividing-radical-expressions\/","title":{"raw":"5.4 Dividing Radical Expressions","rendered":"5.4 Dividing Radical Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Simplify radical expressions by factoring and using the Quotient and\/or Product Rule for Radicals.<\/li>\r\n \t<li>Divide and simplify radical expressions by using the Quotient Rule for Radicals.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe begin with the same premise as the previous section. Can we divide radicals by simply dividing the radicands?\r\n<p style=\"text-align: center;\">[latex]\\dfrac{\\sqrt{18}}{\\sqrt{2}}[\/latex]<\/p>\r\nRemember our philosophy, to look for a corresponding exponent rule that does what we want! The <strong>Power of a Quotient Rule (for exponents)<\/strong> states that [latex] {{\\left( \\dfrac{a}{b} \\right)}^{n}}=\\dfrac{{{a}^{n}}}{{{b}^{n}}}[\/latex]. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: [latex] {{\\left( \\dfrac{a}{b} \\right)}^{\\frac{1}{n}}}=\\dfrac{{{a}^{\\frac{1}{n}}}}{{{b}^{\\frac{1}{n}}}}.[\/latex] Let's use this rule to perform the division in the example above:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\quad\\frac{\\sqrt{18}}{\\sqrt{2}}\\\\\r\n=&amp;\\quad\\frac{18^{1\/2}}{2^{1\/2}}\\\\\r\n=&amp;\\quad\\left(\\frac{18}{2}\\right)^{1\/2} &amp;&amp; \\color{blue}{\\textsf{Power of a Quotient used here}}\\\\\r\n=&amp;\\quad\\sqrt{\\frac{18}{2}}\\\\\r\n=&amp;\\quad 3\\end{align}[\/latex]<\/p>\r\nSimilar to products of radicals, when dividing radicals we divide the radicands if the radicals have the same index. It is crucial that the index on the radicals be the same when we do this or else the exponent rule would not have applied.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>quotient rule for radicals<\/h3>\r\nFor any number [latex]a,[\/latex]any number [latex]b\\neq 0,[\/latex]\u00a0and any positive integer [latex]n[\/latex], if both [latex]\\sqrt[n]{a}[\/latex] and\u00a0[latex]\\sqrt[n]{b}[\/latex] exist then\r\n<p style=\"text-align: center;\">[latex] \\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}=\\sqrt[n]{\\dfrac{a}{b}}[\/latex]<\/p>\r\nRemember that this rule can be applied both forward and backward, just like the product rule.\r\n\r\n<\/div>\r\nAs with multiplication, we will start with some examples featuring integers before moving on to more complex expressions.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSimplify. [latex] \\dfrac{\\sqrt{252}}{\\sqrt{63}}[\/latex]\r\n\r\n[reveal-answer q=\"26879\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"26879\"]\r\n\r\nUse the rule [latex] \\sqrt[n]{\\dfrac{a}{b}}=\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}[\/latex] to rewrite as a single radical. Then divide and simplify.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}\\dfrac{\\sqrt{252}}{\\sqrt{63}}&amp;=\\sqrt{\\dfrac{252}{63}}\\\\\r\n&amp;=\\sqrt{4}\\\\\r\n&amp;=2\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{\\dfrac{48}{25}}[\/latex]\r\n\r\n[reveal-answer q=\"883744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"883744\"]\r\n\r\nUse the rule [latex] \\sqrt[n]{\\dfrac{a}{b}}=\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}[\/latex] to rewrite as two radicals; one in the numerator and one in the denominator.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{\\dfrac{48}{25}}=\\dfrac{\\sqrt{48}}{\\sqrt{25}}[\/latex]<\/p>\r\nSimplify each radical. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;=\\frac{\\sqrt{16\\cdot 3}}{\\sqrt{25}}\\\\\r\n&amp;=\\frac{\\sqrt{4\\cdot 4\\cdot 3}}{\\sqrt{5\\cdot 5}}\\end{align}[\/latex]<\/p>\r\nIdentify and simplify square roots of perfect square factors.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;=\\frac{\\sqrt{(4)^2\\cdot3}}{\\sqrt{(5)^2}}\\\\\r\n&amp;=\\frac{\\sqrt{(4)^2}\\cdot\\sqrt{3}}{\\sqrt{(5)^2}}\\\\\r\n&amp;=\\frac{4\\sqrt{3}}{5}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{\\dfrac{640}{40}}[\/latex]\r\n\r\n[reveal-answer q=\"725564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"725564\"]\r\n\r\nRewrite using the Quotient Rule for Radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{\\dfrac{640}{40}}=\\dfrac{\\sqrt[3]{640}}{\\sqrt[3]{40}}[\/latex]<\/p>\r\nSimplify each radical. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors.\r\n<p style=\"text-align: center;\">[latex] =\\dfrac{\\sqrt[3]{64\\cdot 10}}{\\sqrt[3]{8\\cdot 5}}[\/latex]<\/p>\r\nIdentify and simplify cube roots of perfect cube factors.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}&amp;=\\frac{\\sqrt[3]{{{(4)}^{3}}\\cdot 10}}{\\sqrt[3]{{{(2)}^{3}}\\cdot 5}}\\\\ &amp;=\\frac{\\sqrt[3]{{{(4)}^{3}}}\\cdot \\sqrt[3]{10}}{\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{5}}\\\\\r\n&amp;=\\frac{4\\cdot \\sqrt[3]{10}}{2\\cdot \\sqrt[3]{5}}\\end{align}[\/latex]<\/p>\r\nSimplify this expression even further by combining into a single radical again using the Quotient Rule for Radicals.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;=\\frac{4}{2}\\cdot\\sqrt[3]{\\frac{10}{5}}\\\\\r\n&amp;=2\\sqrt[3]{2}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may have noticed that we could have simplified the radicand in [latex]\\sqrt[3]{\\dfrac{640}{40}}[\/latex] immediately by reducing the fraction prior to applying the Quotient Rule for Radicals. How would this affect the solving process? Let's take another look at the same problem but simplify the radicand first.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{\\dfrac{640}{40}}[\/latex]\r\n\r\n[reveal-answer q=\"403134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"403134\"]\r\n\r\nWithin the radical, divide\u00a0[latex]640[\/latex] by\u00a0[latex]40[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{\\dfrac{640}{40}}=\\sqrt[3]{16}[\/latex]<\/p>\r\nIdentify and simplify cube roots of perfect cube factors.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}&amp;=\\sqrt[3]{{{(2)}^{3}}\\cdot 2}\\\\\r\n&amp;=\\sqrt[3]{{(2)}^{3}}\\cdot\\sqrt[3]{2}\\\\\r\n&amp;=2\\sqrt[3]{2}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice how much more straightforward the second approach was. In general, check to see if you can simplify the expression in the radicand first before applying the Quotient Rule for Radicals.\r\n\r\nIn the next video, we show more examples of simplifying a radical that contains a quotient.\r\n\r\nhttps:\/\/youtu.be\/SxImTm9GVNo\r\n\r\nAs with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide. Whichever order you choose, though, you should arrive at the same final expression.\r\n\r\nNow let us turn to some examples with variables. Notice that the process for dividing these is the same as it is for dividing integers.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\dfrac{\\sqrt{30x}}{\\sqrt{10x}}.[\/latex] Assume all variables represent positive quantities.\r\n\r\n[reveal-answer q=\"236188\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236188\"]\r\n\r\nIn this section we will require variables to be positive (rather than nonnegative) to avoid division by [latex]0[\/latex].\r\n\r\nUse the Quotient Rule for Radicals to rewrite this expression, then simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{\\sqrt{30x}}{\\sqrt{10x}}&amp;=\\sqrt{\\dfrac{30x}{10x}}\\\\\r\n&amp;=\\sqrt{\\dfrac{3\\cdot10x}{10x}}\\\\\r\n&amp;=\\sqrt{3\\cdot\\dfrac{10x}{10x}}\\\\\r\n&amp;=\\sqrt{3\\cdot 1}\\\\\r\n&amp;=\\sqrt{3}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex] \\dfrac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}}.[\/latex] Assume all variables represent positive quantities.\r\n\r\n[reveal-answer q=\"95343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95343\"]\r\n\r\nUse the Quotient Rule for Radicals to rewrite this expression.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}\\dfrac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}}&amp;=\\sqrt[3]{\\dfrac{24x{{y}^{4}}}{8y}}\\\\\r\n&amp;=\\sqrt[3]{\\dfrac{8\\cdot 3\\cdot x\\cdot {{y}^{3}}\\cdot y}{8\\cdot y}}\\\\\r\n&amp;=\\sqrt[3]{\\dfrac{3\\cdot x\\cdot {{y}^{3}}}{1}\\cdot \\dfrac{8y}{8y}}\\\\\r\n&amp;=\\sqrt[3]{3x{{y}^{3}}}\\end{align}[\/latex]<\/p>\r\nIdentify and simplify cube roots of perfect cube factors.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}\r\n&amp;=\\sqrt[3]{{{(y)}^{3}}\\cdot \\,3x}\\\\\r\n&amp;=\\sqrt[3]{(y)^3}\\cdot\\sqrt[3]{3x}\\\\\r\n&amp;=y\\sqrt[3]{3x}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last video, we show more examples of simplifying radicals that contain quotients with variables.\r\n\r\nhttps:\/\/youtu.be\/04X-hMgb0tA\r\n\r\nBe careful to only apply the Quotient Rule for Radicals if the index is the same. For example, while you can think of [latex] \\dfrac{\\sqrt{8{{y}^{2}}}}{\\sqrt{225{{y}^{4}}}}[\/latex] as being equivalent to [latex] \\sqrt{\\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex] since both the numerator and the denominator are square roots, you cannot express [latex] \\dfrac{\\sqrt{8{{y}^{2}}}}{\\sqrt[4]{225{{y}^{4}}}}[\/latex] as [latex] \\sqrt[4]{\\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex] or [latex] \\sqrt{\\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex].\r\n<h2>Summary<\/h2>\r\nThe Quotient Rule for Radicals is similar to the Product Rule and states that [latex] \\sqrt[n]{\\dfrac{a}{b}}=\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}[\/latex]. Both forms (fraction bar inside or outside the radical sign) can be useful for simplifying radical expressions.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Simplify radical expressions by factoring and using the Quotient and\/or Product Rule for Radicals.<\/li>\n<li>Divide and simplify radical expressions by using the Quotient Rule for Radicals.<\/li>\n<\/ul>\n<\/div>\n<p>We begin with the same premise as the previous section. Can we divide radicals by simply dividing the radicands?<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\sqrt{18}}{\\sqrt{2}}[\/latex]<\/p>\n<p>Remember our philosophy, to look for a corresponding exponent rule that does what we want! The <strong>Power of a Quotient Rule (for exponents)<\/strong> states that [latex]{{\\left( \\dfrac{a}{b} \\right)}^{n}}=\\dfrac{{{a}^{n}}}{{{b}^{n}}}[\/latex]. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: [latex]{{\\left( \\dfrac{a}{b} \\right)}^{\\frac{1}{n}}}=\\dfrac{{{a}^{\\frac{1}{n}}}}{{{b}^{\\frac{1}{n}}}}.[\/latex] Let&#8217;s use this rule to perform the division in the example above:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\quad\\frac{\\sqrt{18}}{\\sqrt{2}}\\\\  =&\\quad\\frac{18^{1\/2}}{2^{1\/2}}\\\\  =&\\quad\\left(\\frac{18}{2}\\right)^{1\/2} && \\color{blue}{\\textsf{Power of a Quotient used here}}\\\\  =&\\quad\\sqrt{\\frac{18}{2}}\\\\  =&\\quad 3\\end{align}[\/latex]<\/p>\n<p>Similar to products of radicals, when dividing radicals we divide the radicands if the radicals have the same index. It is crucial that the index on the radicals be the same when we do this or else the exponent rule would not have applied.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>quotient rule for radicals<\/h3>\n<p>For any number [latex]a,[\/latex]any number [latex]b\\neq 0,[\/latex]\u00a0and any positive integer [latex]n[\/latex], if both [latex]\\sqrt[n]{a}[\/latex] and\u00a0[latex]\\sqrt[n]{b}[\/latex] exist then<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}=\\sqrt[n]{\\dfrac{a}{b}}[\/latex]<\/p>\n<p>Remember that this rule can be applied both forward and backward, just like the product rule.<\/p>\n<\/div>\n<p>As with multiplication, we will start with some examples featuring integers before moving on to more complex expressions.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Simplify. [latex]\\dfrac{\\sqrt{252}}{\\sqrt{63}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q26879\">Show Solution<\/span><\/p>\n<div id=\"q26879\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt[n]{\\dfrac{a}{b}}=\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}[\/latex] to rewrite as a single radical. Then divide and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{\\sqrt{252}}{\\sqrt{63}}&=\\sqrt{\\dfrac{252}{63}}\\\\  &=\\sqrt{4}\\\\  &=2\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{\\dfrac{48}{25}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q883744\">Show Solution<\/span><\/p>\n<div id=\"q883744\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt[n]{\\dfrac{a}{b}}=\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}[\/latex] to rewrite as two radicals; one in the numerator and one in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{\\dfrac{48}{25}}=\\dfrac{\\sqrt{48}}{\\sqrt{25}}[\/latex]<\/p>\n<p>Simplify each radical. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&=\\frac{\\sqrt{16\\cdot 3}}{\\sqrt{25}}\\\\  &=\\frac{\\sqrt{4\\cdot 4\\cdot 3}}{\\sqrt{5\\cdot 5}}\\end{align}[\/latex]<\/p>\n<p>Identify and simplify square roots of perfect square factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&=\\frac{\\sqrt{(4)^2\\cdot3}}{\\sqrt{(5)^2}}\\\\  &=\\frac{\\sqrt{(4)^2}\\cdot\\sqrt{3}}{\\sqrt{(5)^2}}\\\\  &=\\frac{4\\sqrt{3}}{5}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{\\dfrac{640}{40}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q725564\">Show Solution<\/span><\/p>\n<div id=\"q725564\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rewrite using the Quotient Rule for Radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{\\dfrac{640}{40}}=\\dfrac{\\sqrt[3]{640}}{\\sqrt[3]{40}}[\/latex]<\/p>\n<p>Simplify each radical. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors.<\/p>\n<p style=\"text-align: center;\">[latex]=\\dfrac{\\sqrt[3]{64\\cdot 10}}{\\sqrt[3]{8\\cdot 5}}[\/latex]<\/p>\n<p>Identify and simplify cube roots of perfect cube factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&=\\frac{\\sqrt[3]{{{(4)}^{3}}\\cdot 10}}{\\sqrt[3]{{{(2)}^{3}}\\cdot 5}}\\\\ &=\\frac{\\sqrt[3]{{{(4)}^{3}}}\\cdot \\sqrt[3]{10}}{\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{5}}\\\\  &=\\frac{4\\cdot \\sqrt[3]{10}}{2\\cdot \\sqrt[3]{5}}\\end{align}[\/latex]<\/p>\n<p>Simplify this expression even further by combining into a single radical again using the Quotient Rule for Radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &=\\frac{4}{2}\\cdot\\sqrt[3]{\\frac{10}{5}}\\\\  &=2\\sqrt[3]{2}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may have noticed that we could have simplified the radicand in [latex]\\sqrt[3]{\\dfrac{640}{40}}[\/latex] immediately by reducing the fraction prior to applying the Quotient Rule for Radicals. How would this affect the solving process? Let&#8217;s take another look at the same problem but simplify the radicand first.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{\\dfrac{640}{40}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q403134\">Show Solution<\/span><\/p>\n<div id=\"q403134\" class=\"hidden-answer\" style=\"display: none\">\n<p>Within the radical, divide\u00a0[latex]640[\/latex] by\u00a0[latex]40[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{\\dfrac{640}{40}}=\\sqrt[3]{16}[\/latex]<\/p>\n<p>Identify and simplify cube roots of perfect cube factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&=\\sqrt[3]{{{(2)}^{3}}\\cdot 2}\\\\  &=\\sqrt[3]{{(2)}^{3}}\\cdot\\sqrt[3]{2}\\\\  &=2\\sqrt[3]{2}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice how much more straightforward the second approach was. In general, check to see if you can simplify the expression in the radicand first before applying the Quotient Rule for Radicals.<\/p>\n<p>In the next video, we show more examples of simplifying a radical that contains a quotient.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Dividing Radicals without Variables (Basic with no rationalizing)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SxImTm9GVNo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>As with multiplication, the main idea here is that sometimes it makes sense to divide and then simplify, and other times it makes sense to simplify and then divide. Whichever order you choose, though, you should arrive at the same final expression.<\/p>\n<p>Now let us turn to some examples with variables. Notice that the process for dividing these is the same as it is for dividing integers.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\dfrac{\\sqrt{30x}}{\\sqrt{10x}}.[\/latex] Assume all variables represent positive quantities.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q236188\">Show Solution<\/span><\/p>\n<div id=\"q236188\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this section we will require variables to be positive (rather than nonnegative) to avoid division by [latex]0[\/latex].<\/p>\n<p>Use the Quotient Rule for Radicals to rewrite this expression, then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{\\sqrt{30x}}{\\sqrt{10x}}&=\\sqrt{\\dfrac{30x}{10x}}\\\\  &=\\sqrt{\\dfrac{3\\cdot10x}{10x}}\\\\  &=\\sqrt{3\\cdot\\dfrac{10x}{10x}}\\\\  &=\\sqrt{3\\cdot 1}\\\\  &=\\sqrt{3}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\dfrac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}}.[\/latex] Assume all variables represent positive quantities.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q95343\">Show Solution<\/span><\/p>\n<div id=\"q95343\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Quotient Rule for Radicals to rewrite this expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\dfrac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}}&=\\sqrt[3]{\\dfrac{24x{{y}^{4}}}{8y}}\\\\  &=\\sqrt[3]{\\dfrac{8\\cdot 3\\cdot x\\cdot {{y}^{3}}\\cdot y}{8\\cdot y}}\\\\  &=\\sqrt[3]{\\dfrac{3\\cdot x\\cdot {{y}^{3}}}{1}\\cdot \\dfrac{8y}{8y}}\\\\  &=\\sqrt[3]{3x{{y}^{3}}}\\end{align}[\/latex]<\/p>\n<p>Identify and simplify cube roots of perfect cube factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}  &=\\sqrt[3]{{{(y)}^{3}}\\cdot \\,3x}\\\\  &=\\sqrt[3]{(y)^3}\\cdot\\sqrt[3]{3x}\\\\  &=y\\sqrt[3]{3x}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last video, we show more examples of simplifying radicals that contain quotients with variables.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Dividing Radicals with Variables (Basic with no rationalizing)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/04X-hMgb0tA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Be careful to only apply the Quotient Rule for Radicals if the index is the same. For example, while you can think of [latex]\\dfrac{\\sqrt{8{{y}^{2}}}}{\\sqrt{225{{y}^{4}}}}[\/latex] as being equivalent to [latex]\\sqrt{\\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex] since both the numerator and the denominator are square roots, you cannot express [latex]\\dfrac{\\sqrt{8{{y}^{2}}}}{\\sqrt[4]{225{{y}^{4}}}}[\/latex] as [latex]\\sqrt[4]{\\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex] or [latex]\\sqrt{\\dfrac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex].<\/p>\n<h2>Summary<\/h2>\n<p>The Quotient Rule for Radicals is similar to the Product Rule and states that [latex]\\sqrt[n]{\\dfrac{a}{b}}=\\dfrac{\\sqrt[n]{a}}{\\sqrt[n]{b}}[\/latex]. Both forms (fraction bar inside or outside the radical sign) can be useful for simplifying radical expressions.<\/p>\n","protected":false},"author":773621,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-642","chapter","type-chapter","status-publish","hentry"],"part":184,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/642","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/users\/773621"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/642\/revisions"}],"predecessor-version":[{"id":1094,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/642\/revisions\/1094"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/parts\/184"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapters\/642\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/media?parent=642"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=642"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/contributor?post=642"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/wp-json\/wp\/v2\/license?post=642"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}