{"id":1653,"date":"2016-06-22T13:22:11","date_gmt":"2016-06-22T13:22:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1653"},"modified":"2019-07-24T21:27:59","modified_gmt":"2019-07-24T21:27:59","slug":"read-or-watch-multiplying-and-dividing-radical-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-or-watch-multiplying-and-dividing-radical-expressions\/","title":{"raw":"Multiply and Divide Radical Expressions","rendered":"Multiply and Divide Radical Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply and divide radical expressions<\/li>\r\n \t<li>Use the product raised to a power rule to multiply radical expressions<\/li>\r\n \t<li>Use the quotient raised to a power rule to divide radical expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nYou can do more than just simplify <strong>radical expressions<\/strong>. You can multiply and divide them, too. The product raised to a power rule that we discussed previously will help us find products of radical expressions. Recall the rule:\r\n<div class=\"textbox shaded\">\r\n<h3>A Product Raised to a Power Rule<\/h3>\r\nFor any numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>: [latex] {{(ab)}^{x}}={{a}^{x}}\\cdot {{b}^{x}}[\/latex]\r\n\r\nFor any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex] {{(ab)}^{\\frac{1}{x}}}={{a}^{\\frac{1}{x}}}\\cdot {{b}^{\\frac{1}{x}}}[\/latex]\r\n\r\nFor any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex] \\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]\r\n\r\n<\/div>\r\nThe Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that you cannot multiply a square root and a cube root using this rule. The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{18}\\cdot \\sqrt{16}[\/latex]\r\n\r\n[reveal-answer q=\"888021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"888021\"]\r\n\r\nUse the rule [latex] \\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{18\\cdot 16}\\\\\\sqrt{288}\\end{array}[\/latex]<\/p>\r\nLook for perfect squares in the radicand, and rewrite the radicand as the product of two factors.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{144\\cdot 2}[\/latex]<\/p>\r\nIdentify perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(12)}^{2}}\\cdot 2}[\/latex]<\/p>\r\nRewrite as the product of two radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(12)}^{2}}}\\cdot \\sqrt{2}[\/latex]<\/p>\r\nSimplify, using [latex] \\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left| 12 \\right|\\cdot \\sqrt{2}\\\\12\\cdot \\sqrt{2}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]12\\sqrt{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may have also noticed that both [latex] \\sqrt{18}[\/latex] and [latex] \\sqrt{16}[\/latex] can be written as products involving perfect square factors. How would the expression change if you simplified each radical first, <i>before<\/i> multiplying? In the next example, we will\u00a0use the same product from above to show that you can simplify before multiplying and get the same result.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{18}\\cdot \\sqrt{16}[\/latex]\r\n\r\n[reveal-answer q=\"479810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"479810\"]\r\n\r\nLook for perfect squares in each radicand, and rewrite as the product of two factors.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}\\sqrt{9\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\\\\\sqrt{3\\cdot 3\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\end{array}[\/latex]<\/p>\r\nIdentify perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(3)}^{2}}\\cdot 2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(3)}^{2}}}\\cdot \\sqrt{2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\r\nSimplify, using [latex] \\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left|3\\right|\\cdot\\sqrt{2}\\cdot\\left|4\\right|\\\\3\\cdot\\sqrt{2}\\cdot4\\end{array}[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]12\\sqrt{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn both cases, you arrive at the same product, [latex] 12\\sqrt{2}[\/latex]. It does not matter whether you multiply the radicands or simplify each radical first.\r\n\r\nYou multiply radical expressions that contain variables in the same manner. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. Look at the two examples that follow. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Note that we specify that the variable is non-negative, [latex] x\\ge 0[\/latex], thus allowing us to avoid the need for absolute value.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{12{{x}^{4}}}\\cdot \\sqrt{3x^2}[\/latex], [latex] x\\ge 0[\/latex]\r\n\r\n[reveal-answer q=\"843487\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843487\"]\r\n\r\nUse the rule [latex] \\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{12{{x}^{4}}\\cdot 3x^2}\\\\\\sqrt{12\\cdot 3\\cdot {{x}^{4}}\\cdot x^2}[\/latex]<\/p>\r\nRecall that [latex] {{x}^{4}}\\cdot x^2={{x}^{4+2}}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{36\\cdot {{x}^{4+2}}}\\\\\\sqrt{36\\cdot {{x}^{6}}}\\end{array}[\/latex]<\/p>\r\nLook for perfect squares in the radicand.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{{{(6)}^{2}}\\cdot {{({{x}^{3}})}^{2}}}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{c}\\sqrt{{{(6)}^{2}}}\\cdot \\sqrt{{{({{x}^{3}})}^{2}}}\\\\6\\cdot {{x}^{3}}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]6{{x}^{3}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Analysis of the Solution<\/h3>\r\nEven though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don't need to write our answer with absolute value because we specified before we simplified that\u00a0[latex] x\\ge 0[\/latex]. It is important to read the problem very well when you are doing math. \u00a0Even the smallest statement like\u00a0[latex] x\\ge 0[\/latex] can influence the way you write your answer.\r\n\r\nIn our next example, we will multiply two cube roots.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{{{x}^{5}}{{y}^{2}}}\\cdot 5\\sqrt[3]{8{{x}^{2}}{{y}^{4}}}[\/latex]\r\n[reveal-answer q=\"399955\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399955\"]\r\n\r\nNotice that <i>both<\/i> radicals are cube roots, so you can use the rule [latex] [\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5\\sqrt[3]{{{x}^{5}}{{y}^{2}}\\cdot 8{{x}^{2}}{{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5}}\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot {{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5+2}}\\cdot {{y}^{2+4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{7}}\\cdot {{y}^{6}}}\\end{array}[\/latex]<\/p>\r\nLook for perfect cubes in the radicand. Since [latex] {{x}^{7}}[\/latex] is not a perfect cube, it has to be rewritten as [latex] {{x}^{6+1}}={{({{x}^{2}})}^{3}}\\cdot x[\/latex].\r\n<p style=\"text-align: center;\">[latex] 5\\sqrt[3]{{{(2)}^{3}}\\cdot {{({{x}^{2}})}^{3}}\\cdot x\\cdot {{({{y}^{2}})}^{3}}}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}5\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{{{({{x}^{2}})}^{3}}}\\cdot \\sqrt[3]{{{({{y}^{2}})}^{3}}}\\cdot \\sqrt[3]{x}\\\\5\\cdot 2\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot \\sqrt[3]{x}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]10{{x}^{2}}{{y}^{2}}\\sqrt[3]{x}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we present more examples of how to multiply radical expressions.\r\n\r\nhttps:\/\/youtu.be\/PQs10_rFrSM\r\n\r\nThis next example is slightly more complicated because there are more than two radicals being multiplied. In this case, notice how the radicals are simplified before multiplication takes place. Remember that the order you choose to use is up to you\u2014you will find that sometimes it is easier to multiply before simplifying, and other times it is easier to simplify before multiplying. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] 2\\sqrt[4]{16{{x}^{9}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{81{{x}^{3}}y}[\/latex], [latex] x\\ge 0[\/latex], [latex] y\\ge 0[\/latex]\r\n\r\n[reveal-answer q=\"257458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"257458\"]\r\n\r\nNotice this expression is multiplying three radicals with the same (fourth) root. Simplify each radical, if possible, before multiplying. Be looking for powers of\u00a0[latex]4[\/latex] in each radicand.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[4]{{{(2)}^{4}}\\cdot {{({{x}^{2}})}^{4}}\\cdot x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}\\cdot {{x}^{3}}y}[\/latex]<\/p>\r\nRewrite as the product of radicals.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[4]{{{(2)}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}}\\cdot \\sqrt[4]{{{x}^{3}}y}[\/latex]<\/p>\r\nIdentify and pull out powers of\u00a0[latex]4[\/latex], using the fact that [latex] \\sqrt[4]{{{x}^{4}}}=\\left| x \\right|[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}2\\cdot \\left| 2 \\right|\\cdot \\left| {{x}^{2}} \\right|\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\left| 3 \\right|\\cdot \\sqrt[4]{{{x}^{3}}y}\\\\2\\cdot 2\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot 3\\cdot \\sqrt[4]{{{x}^{3}}y}\\end{array}[\/latex]<\/p>\r\nSince all the radicals are fourth roots, you can use the rule [latex] \\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex] to multiply the radicands.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot 2\\cdot 3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x\\cdot {{y}^{3}}\\cdot {{x}^{3}}y}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{1+3}}\\cdot {{y}^{3+1}}}\\end{array}[\/latex]<\/p>\r\nNow that the radicands have been multiplied, look again for powers of\u00a0[latex]4[\/latex], and pull them out. We can drop the absolute value signs in our final answer because at the start of the problem we were told [latex] x\\ge 0[\/latex], [latex] y\\ge 0[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{l}12{{x}^{2}}\\sqrt[4]{{{x}^{4}}\\cdot {{y}^{4}}}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{4}}}\\cdot \\sqrt[4]{{{y}^{4}}}\\\\12{{x}^{2}}\\cdot \\left| x \\right|\\cdot \\left| y \\right|\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]12{{x}^{3}}y,\\,\\,x\\ge 0,\\,\\,y\\ge 0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of multiplying cube roots.\r\n\r\nhttps:\/\/youtu.be\/cxRXofdelIM\r\n<h2>Dividing Radical Expressions<\/h2>\r\nYou can use the same ideas to help you figure out how to simplify and divide radical expressions. Recall that the Product Raised to a Power Rule states that [latex] \\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]. Well, what if you are dealing with a quotient instead of a product?\r\n\r\nThere is a rule for that, too. The <strong>Quotient Raised to a Power Rule<\/strong> states that [latex] {{\\left( \\frac{a}{b} \\right)}^{x}}=\\frac{{{a}^{x}}}{{{b}^{x}}}[\/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( \\frac{a}{b} \\right)}^{\\frac{1}{x}}}=\\frac{{{a}^{\\frac{1}{x}}}}{{{b}^{\\frac{1}{x}}}}[\/latex], so [latex] \\sqrt[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>A Quotient Raised to a Power Rule<\/h3>\r\nFor any real numbers <i>a<\/i> and <i>b<\/i> (<i>b<\/i> \u2260 0) and any positive integer <i>x<\/i>: [latex] {{\\left( \\frac{a}{b} \\right)}^{\\frac{1}{x}}}=\\frac{{{a}^{\\frac{1}{x}}}}{{{b}^{\\frac{1}{x}}}}[\/latex]\r\n\r\nFor any real numbers <i>a<\/i> and <i>b<\/i> (<i>b<\/i> \u2260 0) and any positive integer <i>x<\/i>: [latex] \\sqrt[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex]\r\n\r\n<\/div>\r\nAs you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like [latex] \\frac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}}[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt{\\frac{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[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex] to create two radicals; one in the numerator and one in the denominator.\r\n<p style=\"text-align: center;\">[latex] \\frac{\\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{array}{c}\\frac{\\sqrt{16\\cdot 3}}{\\sqrt{25}}\\\\\\\\\\text{or}\\\\\\\\\\frac{\\sqrt{4\\cdot 4\\cdot 3}}{\\sqrt{5\\cdot 5}}\\end{array}[\/latex]<\/p>\r\nIdentify and pull out perfect squares.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}\\frac{\\sqrt{{{(4)}^{2}}\\cdot 3}}{\\sqrt{{{(5)}^{2}}}}\\\\\\\\\\frac{\\sqrt{{{(4)}^{2}}}\\cdot \\sqrt{3}}{\\sqrt{{{(5)}^{2}}}}\\end{array}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\frac{4\\cdot \\sqrt{3}}{5}[\/latex]<\/p>\r\nThe answer is [latex]\\frac{4\\sqrt{3}}{5}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\sqrt[3]{\\frac{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 Raised to a Power Rule.\r\n<p style=\"text-align: center;\">[latex] \\frac{\\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] \\frac{\\sqrt[3]{64\\cdot 10}}{\\sqrt[3]{8\\cdot 5}}[\/latex]<\/p>\r\nIdentify and pull out perfect cubes.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}\\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{array}[\/latex]<\/p>\r\nYou can simplify this expression even further by looking for common factors in the numerator and denominator.\r\n<p style=\"text-align: center;\">[latex] \\frac{4\\sqrt[3]{10}}{2\\sqrt[3]{5}}[\/latex]<\/p>\r\nRewrite the numerator as a product of factors.\r\n<p style=\"text-align: center;\">[latex] \\frac{2\\cdot 2\\sqrt[3]{5}\\cdot \\sqrt[3]{2}}{2\\sqrt[3]{5}}[\/latex]<\/p>\r\nIdentify factors of\u00a0[latex]1[\/latex], and simplify.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}2\\cdot \\frac{2\\sqrt[3]{5}}{2\\sqrt[3]{5}}\\cdot \\sqrt[3]{2}\\\\\\\\2\\cdot 1\\cdot \\sqrt[3]{2}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]2\\sqrt[3]{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThat was a lot of effort, but you were able to simplify using the Quotient Raised to a Power Rule. What if you found the quotient of this expression by dividing within the radical first and then took the cube root of the quotient?\r\n\r\nNow take another look at that problem using this approach.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\frac{\\sqrt[3]{640}}{\\sqrt[3]{40}}[\/latex]\r\n\r\n[reveal-answer q=\"403134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"403134\"]\r\n\r\nSince both radicals are cube roots, you can use the rule [latex] \\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}=\\sqrt[x]{\\frac{a}{b}}[\/latex] to create a single rational expression underneath the radical.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{\\frac{640}{40}}[\/latex]<\/p>\r\nWithin the radical, divide\u00a0[latex]640[\/latex] by\u00a0[latex]40[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}640\\div 40=16\\\\\\sqrt[3]{16}\\end{array}[\/latex]<\/p>\r\nLook for perfect cubes in the radicand, and rewrite the radicand as a product of factors.\r\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{8\\cdot2}[\/latex]<\/p>\r\nIdentify perfect cubes and pull them out.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}\\sqrt[3]{{{(2)}^{3}}\\cdot 2}\\\\\\sqrt[3]{{(2)}^{3}}\\cdot\\sqrt[3]{2}\\end{array}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]2\\cdot\\sqrt[3]{2}[\/latex]<\/p>\r\nThe answer is [latex]2\\sqrt[3]{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice how much more straightforward the approach was.\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 radical expressions containing division. 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]\\frac{\\sqrt{30x}}{\\sqrt{10x}},x&gt;0[\/latex]\r\n\r\n[reveal-answer q=\"236188\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"236188\"]\r\n\r\nUse the Quotient Raised to a Power Rule to rewrite this expression.\r\n<p style=\"text-align: center;\">[latex]\\sqrt{\\frac{30x}{10x}}[\/latex]<\/p>\r\nSimplify [latex] \\sqrt{\\frac{30x}{10x}}[\/latex] by identifying similar factors in the numerator and denominator and then identifying factors of\u00a0[latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{\\frac{3\\cdot10x}{10x}}\\\\\\\\\\sqrt{3\\cdot\\frac{10x}{10x}}\\\\\\\\\\sqrt{3\\cdot1}\\end{array}[\/latex]<\/p>\r\nThe answer is [latex]\\sqrt{3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\frac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}},\\,\\,y\\ne 0[\/latex]\r\n\r\n[reveal-answer q=\"95343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95343\"]\r\n\r\nUse the Quotient Raised to a Power Rule to rewrite this expression.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{\\frac{24x{{y}^{4}}}{8y}}[\/latex]<\/p>\r\nSimplify [latex] \\sqrt[3]{\\frac{24x{{y}^{4}}}{8y}}[\/latex] by identifying similar factors in the numerator and denominator and then identifying factors of\u00a0[latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\sqrt[3]{\\frac{8\\cdot 3\\cdot x\\cdot {{y}^{3}}\\cdot y}{8\\cdot y}}\\\\\\\\\\sqrt[3]{\\frac{3\\cdot x\\cdot {{y}^{3}}}{1}\\cdot \\frac{8y}{8y}}\\\\\\\\\\sqrt[3]{\\frac{3\\cdot x\\cdot {{y}^{3}}}{1}\\cdot 1}\\end{array}[\/latex]<\/p>\r\nIdentify perfect cubes and pull them out of the radical.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{3x{{y}^{3}}}\\\\\\sqrt[3]{{{(y)}^{3}}\\cdot \\,3x}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{{{(y)}^{3}}}\\cdot \\,\\sqrt[3]{3x}[\/latex]<\/p>\r\nThe answer is [latex]y\\,\\sqrt[3]{3x}[\/latex].\r\n\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\nAs you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. For example, while you can think of [latex] \\frac{\\sqrt{8{{y}^{2}}}}{\\sqrt{225{{y}^{4}}}}[\/latex] as being equivalent to [latex] \\sqrt{\\frac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex] since both the numerator and the denominator are square roots, notice that you cannot express [latex] \\frac{\\sqrt{8{{y}^{2}}}}{\\sqrt[4]{225{{y}^{4}}}}[\/latex] as [latex] \\sqrt[4]{\\frac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex]. In this second case, the numerator is a square root and the denominator is a fourth root.\r\n<h2>Summary<\/h2>\r\nThe Product Raised to a Power Rule and the Quotient Raised to a Power Rule can be used to simplify radical expressions as long as the roots of the radicals are the same. The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The same is true of roots: [latex] \\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]. When dividing radical expressions, the rules governing quotients are similar: [latex] \\sqrt[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply and divide radical expressions<\/li>\n<li>Use the product raised to a power rule to multiply radical expressions<\/li>\n<li>Use the quotient raised to a power rule to divide radical expressions<\/li>\n<\/ul>\n<\/div>\n<p>You can do more than just simplify <strong>radical expressions<\/strong>. You can multiply and divide them, too. The product raised to a power rule that we discussed previously will help us find products of radical expressions. Recall the rule:<\/p>\n<div class=\"textbox shaded\">\n<h3>A Product Raised to a Power Rule<\/h3>\n<p>For any numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>: [latex]{{(ab)}^{x}}={{a}^{x}}\\cdot {{b}^{x}}[\/latex]<\/p>\n<p>For any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex]{{(ab)}^{\\frac{1}{x}}}={{a}^{\\frac{1}{x}}}\\cdot {{b}^{\\frac{1}{x}}}[\/latex]<\/p>\n<p>For any numbers <i>a<\/i> and <i>b<\/i> and any positive integer <i>x<\/i>: [latex]\\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]<\/p>\n<\/div>\n<p>The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that you cannot multiply a square root and a cube root using this rule. The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{18}\\cdot \\sqrt{16}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q888021\">Show Solution<\/span><\/p>\n<div id=\"q888021\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{18\\cdot 16}\\\\\\sqrt{288}\\end{array}[\/latex]<\/p>\n<p>Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{144\\cdot 2}[\/latex]<\/p>\n<p>Identify perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(12)}^{2}}\\cdot 2}[\/latex]<\/p>\n<p>Rewrite as the product of two radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(12)}^{2}}}\\cdot \\sqrt{2}[\/latex]<\/p>\n<p>Simplify, using [latex]\\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\left| 12 \\right|\\cdot \\sqrt{2}\\\\12\\cdot \\sqrt{2}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]12\\sqrt{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may have also noticed that both [latex]\\sqrt{18}[\/latex] and [latex]\\sqrt{16}[\/latex] can be written as products involving perfect square factors. How would the expression change if you simplified each radical first, <i>before<\/i> multiplying? In the next example, we will\u00a0use the same product from above to show that you can simplify before multiplying and get the same result.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{18}\\cdot \\sqrt{16}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q479810\">Show Solution<\/span><\/p>\n<div id=\"q479810\" class=\"hidden-answer\" style=\"display: none\">\n<p>Look for perfect squares in each radicand, and rewrite as the product of two factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{9\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\\\\\sqrt{3\\cdot 3\\cdot 2}\\cdot \\sqrt{4\\cdot 4}\\end{array}[\/latex]<\/p>\n<p>Identify perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(3)}^{2}}\\cdot 2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(3)}^{2}}}\\cdot \\sqrt{2}\\cdot \\sqrt{{{(4)}^{2}}}[\/latex]<\/p>\n<p>Simplify, using [latex]\\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left|3\\right|\\cdot\\sqrt{2}\\cdot\\left|4\\right|\\\\3\\cdot\\sqrt{2}\\cdot4\\end{array}[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]12\\sqrt{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In both cases, you arrive at the same product, [latex]12\\sqrt{2}[\/latex]. It does not matter whether you multiply the radicands or simplify each radical first.<\/p>\n<p>You multiply radical expressions that contain variables in the same manner. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. Look at the two examples that follow. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Note that we specify that the variable is non-negative, [latex]x\\ge 0[\/latex], thus allowing us to avoid the need for absolute value.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{12{{x}^{4}}}\\cdot \\sqrt{3x^2}[\/latex], [latex]x\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q843487\">Show Solution<\/span><\/p>\n<div id=\"q843487\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the rule [latex]\\sqrt[x]{a}\\cdot \\sqrt[x]{b}=\\sqrt[x]{ab}[\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{12{{x}^{4}}\\cdot 3x^2}\\\\\\sqrt{12\\cdot 3\\cdot {{x}^{4}}\\cdot x^2}[\/latex]<\/p>\n<p>Recall that [latex]{{x}^{4}}\\cdot x^2={{x}^{4+2}}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{36\\cdot {{x}^{4+2}}}\\\\\\sqrt{36\\cdot {{x}^{6}}}\\end{array}[\/latex]<\/p>\n<p>Look for perfect squares in the radicand.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{{{(6)}^{2}}\\cdot {{({{x}^{3}})}^{2}}}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\sqrt{{{(6)}^{2}}}\\cdot \\sqrt{{{({{x}^{3}})}^{2}}}\\\\6\\cdot {{x}^{3}}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]6{{x}^{3}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\n<p>Even though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don&#8217;t need to write our answer with absolute value because we specified before we simplified that\u00a0[latex]x\\ge 0[\/latex]. It is important to read the problem very well when you are doing math. \u00a0Even the smallest statement like\u00a0[latex]x\\ge 0[\/latex] can influence the way you write your answer.<\/p>\n<p>In our next example, we will multiply two cube roots.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt[3]{{{x}^{5}}{{y}^{2}}}\\cdot 5\\sqrt[3]{8{{x}^{2}}{{y}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q399955\">Show Solution<\/span><\/p>\n<div id=\"q399955\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice that <i>both<\/i> radicals are cube roots, so you can use the rule [latex][\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5\\sqrt[3]{{{x}^{5}}{{y}^{2}}\\cdot 8{{x}^{2}}{{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5}}\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot {{y}^{4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{5+2}}\\cdot {{y}^{2+4}}}\\\\5\\sqrt[3]{8\\cdot {{x}^{7}}\\cdot {{y}^{6}}}\\end{array}[\/latex]<\/p>\n<p>Look for perfect cubes in the radicand. Since [latex]{{x}^{7}}[\/latex] is not a perfect cube, it has to be rewritten as [latex]{{x}^{6+1}}={{({{x}^{2}})}^{3}}\\cdot x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]5\\sqrt[3]{{{(2)}^{3}}\\cdot {{({{x}^{2}})}^{3}}\\cdot x\\cdot {{({{y}^{2}})}^{3}}}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}5\\sqrt[3]{{{(2)}^{3}}}\\cdot \\sqrt[3]{{{({{x}^{2}})}^{3}}}\\cdot \\sqrt[3]{{{({{y}^{2}})}^{3}}}\\cdot \\sqrt[3]{x}\\\\5\\cdot 2\\cdot {{x}^{2}}\\cdot {{y}^{2}}\\cdot \\sqrt[3]{x}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]10{{x}^{2}}{{y}^{2}}\\sqrt[3]{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we present more examples of how to multiply radical expressions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiply Square Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/PQs10_rFrSM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>This next example is slightly more complicated because there are more than two radicals being multiplied. In this case, notice how the radicals are simplified before multiplication takes place. Remember that the order you choose to use is up to you\u2014you will find that sometimes it is easier to multiply before simplifying, and other times it is easier to simplify before multiplying. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]2\\sqrt[4]{16{{x}^{9}}}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{81{{x}^{3}}y}[\/latex], [latex]x\\ge 0[\/latex], [latex]y\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q257458\">Show Solution<\/span><\/p>\n<div id=\"q257458\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice this expression is multiplying three radicals with the same (fourth) root. Simplify each radical, if possible, before multiplying. Be looking for powers of\u00a0[latex]4[\/latex] in each radicand.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[4]{{{(2)}^{4}}\\cdot {{({{x}^{2}})}^{4}}\\cdot x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}\\cdot {{x}^{3}}y}[\/latex]<\/p>\n<p>Rewrite as the product of radicals.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[4]{{{(2)}^{4}}}\\cdot \\sqrt[4]{{{({{x}^{2}})}^{4}}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\sqrt[4]{{{(3)}^{4}}}\\cdot \\sqrt[4]{{{x}^{3}}y}[\/latex]<\/p>\n<p>Identify and pull out powers of\u00a0[latex]4[\/latex], using the fact that [latex]\\sqrt[4]{{{x}^{4}}}=\\left| x \\right|[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot \\left| 2 \\right|\\cdot \\left| {{x}^{2}} \\right|\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot \\left| 3 \\right|\\cdot \\sqrt[4]{{{x}^{3}}y}\\\\2\\cdot 2\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x}\\cdot \\sqrt[4]{{{y}^{3}}}\\cdot 3\\cdot \\sqrt[4]{{{x}^{3}}y}\\end{array}[\/latex]<\/p>\n<p>Since all the radicals are fourth roots, you can use the rule [latex]\\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex] to multiply the radicands.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot 2\\cdot 3\\cdot {{x}^{2}}\\cdot \\sqrt[4]{x\\cdot {{y}^{3}}\\cdot {{x}^{3}}y}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{1+3}}\\cdot {{y}^{3+1}}}\\end{array}[\/latex]<\/p>\n<p>Now that the radicands have been multiplied, look again for powers of\u00a0[latex]4[\/latex], and pull them out. We can drop the absolute value signs in our final answer because at the start of the problem we were told [latex]x\\ge 0[\/latex], [latex]y\\ge 0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}12{{x}^{2}}\\sqrt[4]{{{x}^{4}}\\cdot {{y}^{4}}}\\\\12{{x}^{2}}\\sqrt[4]{{{x}^{4}}}\\cdot \\sqrt[4]{{{y}^{4}}}\\\\12{{x}^{2}}\\cdot \\left| x \\right|\\cdot \\left| y \\right|\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]12{{x}^{3}}y,\\,\\,x\\ge 0,\\,\\,y\\ge 0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of multiplying cube roots.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Multiply Cube Roots\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/cxRXofdelIM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Dividing Radical Expressions<\/h2>\n<p>You can use the same ideas to help you figure out how to simplify and divide radical expressions. Recall that the Product Raised to a Power Rule states that [latex]\\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]. Well, what if you are dealing with a quotient instead of a product?<\/p>\n<p>There is a rule for that, too. The <strong>Quotient Raised to a Power Rule<\/strong> states that [latex]{{\\left( \\frac{a}{b} \\right)}^{x}}=\\frac{{{a}^{x}}}{{{b}^{x}}}[\/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( \\frac{a}{b} \\right)}^{\\frac{1}{x}}}=\\frac{{{a}^{\\frac{1}{x}}}}{{{b}^{\\frac{1}{x}}}}[\/latex], so [latex]\\sqrt[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>A Quotient Raised to a Power Rule<\/h3>\n<p>For any real numbers <i>a<\/i> and <i>b<\/i> (<i>b<\/i> \u2260 0) and any positive integer <i>x<\/i>: [latex]{{\\left( \\frac{a}{b} \\right)}^{\\frac{1}{x}}}=\\frac{{{a}^{\\frac{1}{x}}}}{{{b}^{\\frac{1}{x}}}}[\/latex]<\/p>\n<p>For any real numbers <i>a<\/i> and <i>b<\/i> (<i>b<\/i> \u2260 0) and any positive integer <i>x<\/i>: [latex]\\sqrt[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex]<\/p>\n<\/div>\n<p>As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like [latex]\\frac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}}[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{\\frac{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[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex] to create two radicals; one in the numerator and one in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\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{array}{c}\\frac{\\sqrt{16\\cdot 3}}{\\sqrt{25}}\\\\\\\\\\text{or}\\\\\\\\\\frac{\\sqrt{4\\cdot 4\\cdot 3}}{\\sqrt{5\\cdot 5}}\\end{array}[\/latex]<\/p>\n<p>Identify and pull out perfect squares.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{\\sqrt{{{(4)}^{2}}\\cdot 3}}{\\sqrt{{{(5)}^{2}}}}\\\\\\\\\\frac{\\sqrt{{{(4)}^{2}}}\\cdot \\sqrt{3}}{\\sqrt{{{(5)}^{2}}}}\\end{array}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4\\cdot \\sqrt{3}}{5}[\/latex]<\/p>\n<p>The answer is [latex]\\frac{4\\sqrt{3}}{5}[\/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]{\\frac{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 Raised to a Power Rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\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]\\frac{\\sqrt[3]{64\\cdot 10}}{\\sqrt[3]{8\\cdot 5}}[\/latex]<\/p>\n<p>Identify and pull out perfect cubes.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\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{array}[\/latex]<\/p>\n<p>You can simplify this expression even further by looking for common factors in the numerator and denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4\\sqrt[3]{10}}{2\\sqrt[3]{5}}[\/latex]<\/p>\n<p>Rewrite the numerator as a product of factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2\\cdot 2\\sqrt[3]{5}\\cdot \\sqrt[3]{2}}{2\\sqrt[3]{5}}[\/latex]<\/p>\n<p>Identify factors of\u00a0[latex]1[\/latex], and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2\\cdot \\frac{2\\sqrt[3]{5}}{2\\sqrt[3]{5}}\\cdot \\sqrt[3]{2}\\\\\\\\2\\cdot 1\\cdot \\sqrt[3]{2}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]2\\sqrt[3]{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>That was a lot of effort, but you were able to simplify using the Quotient Raised to a Power Rule. What if you found the quotient of this expression by dividing within the radical first and then took the cube root of the quotient?<\/p>\n<p>Now take another look at that problem using this approach.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\frac{\\sqrt[3]{640}}{\\sqrt[3]{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>Since both radicals are cube roots, you can use the rule [latex]\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}=\\sqrt[x]{\\frac{a}{b}}[\/latex] to create a single rational expression underneath the radical.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{\\frac{640}{40}}[\/latex]<\/p>\n<p>Within the radical, divide\u00a0[latex]640[\/latex] by\u00a0[latex]40[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}640\\div 40=16\\\\\\sqrt[3]{16}\\end{array}[\/latex]<\/p>\n<p>Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{8\\cdot2}[\/latex]<\/p>\n<p>Identify perfect cubes and pull them out.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt[3]{{{(2)}^{3}}\\cdot 2}\\\\\\sqrt[3]{{(2)}^{3}}\\cdot\\sqrt[3]{2}\\end{array}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot\\sqrt[3]{2}[\/latex]<\/p>\n<p>The answer is [latex]2\\sqrt[3]{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice how much more straightforward the approach was.<\/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-3\" 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 radical expressions containing division. 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]\\frac{\\sqrt{30x}}{\\sqrt{10x}},x>0[\/latex]<\/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>Use the Quotient Raised to a Power Rule to rewrite this expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{\\frac{30x}{10x}}[\/latex]<\/p>\n<p>Simplify [latex]\\sqrt{\\frac{30x}{10x}}[\/latex] by identifying similar factors in the numerator and denominator and then identifying factors of\u00a0[latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\sqrt{\\frac{3\\cdot10x}{10x}}\\\\\\\\\\sqrt{3\\cdot\\frac{10x}{10x}}\\\\\\\\\\sqrt{3\\cdot1}\\end{array}[\/latex]<\/p>\n<p>The answer is [latex]\\sqrt{3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\frac{\\sqrt[3]{24x{{y}^{4}}}}{\\sqrt[3]{8y}},\\,\\,y\\ne 0[\/latex]<\/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 Raised to a Power Rule to rewrite this expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{\\frac{24x{{y}^{4}}}{8y}}[\/latex]<\/p>\n<p>Simplify [latex]\\sqrt[3]{\\frac{24x{{y}^{4}}}{8y}}[\/latex] by identifying similar factors in the numerator and denominator and then identifying factors of\u00a0[latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\sqrt[3]{\\frac{8\\cdot 3\\cdot x\\cdot {{y}^{3}}\\cdot y}{8\\cdot y}}\\\\\\\\\\sqrt[3]{\\frac{3\\cdot x\\cdot {{y}^{3}}}{1}\\cdot \\frac{8y}{8y}}\\\\\\\\\\sqrt[3]{\\frac{3\\cdot x\\cdot {{y}^{3}}}{1}\\cdot 1}\\end{array}[\/latex]<\/p>\n<p>Identify perfect cubes and pull them out of the radical.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{3x{{y}^{3}}}\\\\\\sqrt[3]{{{(y)}^{3}}\\cdot \\,3x}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{{(y)}^{3}}}\\cdot \\,\\sqrt[3]{3x}[\/latex]<\/p>\n<p>The answer is [latex]y\\,\\sqrt[3]{3x}[\/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-4\" 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>As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. For example, while you can think of [latex]\\frac{\\sqrt{8{{y}^{2}}}}{\\sqrt{225{{y}^{4}}}}[\/latex] as being equivalent to [latex]\\sqrt{\\frac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex] since both the numerator and the denominator are square roots, notice that you cannot express [latex]\\frac{\\sqrt{8{{y}^{2}}}}{\\sqrt[4]{225{{y}^{4}}}}[\/latex] as [latex]\\sqrt[4]{\\frac{8{{y}^{2}}}{225{{y}^{4}}}}[\/latex]. In this second case, the numerator is a square root and the denominator is a fourth root.<\/p>\n<h2>Summary<\/h2>\n<p>The Product Raised to a Power Rule and the Quotient Raised to a Power Rule can be used to simplify radical expressions as long as the roots of the radicals are the same. The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The same is true of roots: [latex]\\sqrt[x]{ab}=\\sqrt[x]{a}\\cdot \\sqrt[x]{b}[\/latex]. When dividing radical expressions, the rules governing quotients are similar: [latex]\\sqrt[x]{\\frac{a}{b}}=\\frac{\\sqrt[x]{a}}{\\sqrt[x]{b}}[\/latex].<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1653\">\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>Multiply Square Roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/PQs10_rFrSM\">https:\/\/youtu.be\/PQs10_rFrSM<\/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>Multiply Cube Roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/cxRXofdelIM\">https:\/\/youtu.be\/cxRXofdelIM<\/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>Dividing Radicals without Variables (Basic with no rationalizing). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/SxImTm9GVNo\">https:\/\/youtu.be\/SxImTm9GVNo<\/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>Dividing Radicals with Variables (Basic with no rationalizing). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/04X-hMgb0tA\">https:\/\/youtu.be\/04X-hMgb0tA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Authored by<\/strong>: Abramson, Jay. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Dwonload fro free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Multiply Square Roots\",\"author\":\"James Sousa 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