{"id":1657,"date":"2016-06-22T13:23:45","date_gmt":"2016-06-22T13:23:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1657"},"modified":"2019-07-24T21:28:17","modified_gmt":"2019-07-24T21:28:17","slug":"read-or-watch-multiplication-of-multiple-term-radicals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-or-watch-multiplication-of-multiple-term-radicals\/","title":{"raw":"Multiple Term Radicals","rendered":"Multiple Term Radicals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Use the distributive property to multiply multiple term radicals and then simplify<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen multiplying multiple term radical expressions, it is important to follow the <strong>Distributive Property of Multiplication<\/strong>, as when you are multiplying regular, non-radical expressions.\r\n\r\nRadicals follow the same mathematical rules that other real numbers do. So, although the expression [latex] \\sqrt{x}(3\\sqrt{x}-5)[\/latex] may look different than [latex] a(3a-5)[\/latex], you can treat them the same way.\r\n\r\nLet us have a look at how to apply the Distributive Property. First let us do a problem with the variable <i>a<\/i>, and then solve the same problem replacing <i>a<\/i> with [latex] \\sqrt{x}[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] a(3a-5)[\/latex]\r\n\r\n[reveal-answer q=\"653719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"653719\"]\r\n\r\nUse the Distributive Property of Multiplication over Subtraction.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}a(3a)-a(5)\\\\=3a^2-5a\\end{array}[\/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{x}(3\\sqrt{x}-5)[\/latex]\r\n\r\n[reveal-answer q=\"886472\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"886472\"]\r\n\r\nUse the Distributive Property of Multiplication over Subtraction.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{x}(3\\sqrt{x})-\\sqrt{x}(5)[\/latex]<\/p>\r\nApply the rules of multiplying radicals: [latex] \\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex] to multiply [latex] \\sqrt{x}(3\\sqrt{x})[\/latex].\r\n<p style=\"text-align: center;\">[latex] 3\\sqrt{{{x}^{2}}}-5\\sqrt{x}[\/latex]<\/p>\r\nBe sure to simplify radicals when you can: [latex] \\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex], so [latex] 3\\sqrt{{{x}^{2}}}=3\\left| x \\right|[\/latex].\r\n\r\nThe answer is [latex]3\\left| x \\right|-5\\sqrt{x}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe answers to the previous two problems should look similar to you. The only difference is that in the second problem, [latex] \\sqrt{x}[\/latex] has replaced the variable <i>a <\/i>(and so [latex] \\left| x \\right|[\/latex] has replaced <i>a<\/i><sup>2<\/sup>). The process of multiplying is very much the same in both problems.\r\n\r\nIn these next two problems, each term contains a radical.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] 7\\sqrt{x}\\left( 2\\sqrt{xy}+\\sqrt{y} \\right)[\/latex]\r\n\r\n[reveal-answer q=\"732671\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"732671\"]\r\n\r\nUse the Distributive Property of Multiplication over Addition to multiply each term within parentheses by [latex] 7\\sqrt{x}[\/latex].\r\n<p style=\"text-align: center;\">[latex] 7\\sqrt{x}\\left( 2\\sqrt{xy} \\right)+7\\sqrt{x}\\left( \\sqrt{y} \\right)[\/latex]<\/p>\r\nApply the rules of multiplying radicals.\r\n<p style=\"text-align: center;\">[latex] 7\\cdot 2\\sqrt{{{x}^{2}}y}+7\\sqrt{xy}[\/latex]<\/p>\r\n[latex] \\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex], so [latex] \\left| x \\right|[\/latex] can be pulled out of the radical.\r\n<p style=\"text-align: center;\">[latex] 14|x|\\sqrt{y}+7\\sqrt{xy}[\/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]{a}\\left( 2\\sqrt[3]{{{a}^{2}}}-4\\sqrt[3]{{{a}^{5}}}+8\\sqrt[3]{{{a}^{8}}} \\right)[\/latex]\r\n\r\n[reveal-answer q=\"100802\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"100802\"]\r\n\r\nUse the Distributive Property.\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{a}\\left( 2\\sqrt[3]{{{a}^{2}}} \\right)-\\sqrt[3]{a}\\left( 4\\sqrt[3]{{{a}^{5}}} \\right)+\\sqrt[3]{a}\\left( 8\\sqrt[3]{{{a}^{8}}} \\right)[\/latex]<\/p>\r\nApply the rules of multiplying radicals.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{c}2\\sqrt[3]{a\\cdot {{a}^{2}}}-4\\sqrt[3]{a\\cdot {{a}^{5}}}+8\\sqrt[3]{a\\cdot {{a}^{8}}}\\\\2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{a}^{6}}}+8\\sqrt[3]{{{a}^{9}}}\\end{array}[\/latex]<\/p>\r\nIdentify cubes in each of the radicals.\r\n<p style=\"text-align: center;\">[latex] 2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{\\left( {{a}^{2}} \\right)}^{3}}}+8\\sqrt[3]{{{\\left( {{a}^{3}} \\right)}^{3}}}[\/latex]<\/p>\r\nThe solution is [latex]2a-4{{a}^{2}}+8{{a}^{3}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of how to multiply radical expressions using the distributive property.\r\n\r\nhttps:\/\/youtu.be\/hizqmgBjW0k\r\n\r\nIn all of these examples, multiplication of radicals has been shown following the pattern [latex] \\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex]. Then, only after multiplying, some radicals have been simplified\u2014like in the last problem. After you have worked with radical expressions a bit more, you may feel more comfortable identifying quantities such as [latex] \\sqrt{x}\\cdot \\sqrt{x}=x[\/latex] without going through the intermediate step of finding that [latex] \\sqrt{x}\\cdot \\sqrt{x}=\\sqrt{{{x}^{2}}}[\/latex]. In the rest of the examples that follow, though, each step is shown.\r\n<h2>Multiply Binomial Expressions That Contain Radicals<\/h2>\r\nYou can use the same technique for multiplying binomials to multiply binomial\u00a0expressions with radicals.\r\n\r\nAs a refresher, here is the process for multiplying two binomials. If you like using the expression \u201cFOIL\u201d (First, Outside, Inside, Last) to help you figure out the order in which the terms should be multiplied, you can use it here, too.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. [latex] \\left( 2x+5 \\right)\\left( 3x-2 \\right)[\/latex]\r\n\r\n[reveal-answer q=\"577475\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"577475\"]\r\n\r\nUse the Distributive Property.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x\\cdot 3x=6{{x}^{2}}\\\\\\text{Outside}:\\,\\,\\,2x\\cdot \\left( -2 \\right)=-4x\\\\\\text{Inside}:\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot 3x=15x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot \\left( -2 \\right)=-10\\end{array}[\/latex]<\/p>\r\nRecord the terms, and then combine like terms.\r\n<p style=\"text-align: center;\">[latex] 6{{x}^{2}}-4x+15x-10[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]6{{x}^{2}}+11x-10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere is the same problem, with [latex] \\sqrt{b}[\/latex] replacing the variable <i>x<\/i>.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. [latex] \\left( 2\\sqrt{b}+5 \\right)\\left( 3\\sqrt{b}-2 \\right),\\,\\,b\\ge 0[\/latex]\r\n\r\n[reveal-answer q=\"674608\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"674608\"]\r\n\r\nUse the Distributive Property to multiply. Simplify using [latex] \\sqrt{x}\\cdot \\sqrt{x}=x[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\sqrt{b}\\cdot 3\\sqrt{b}=2\\cdot 3\\cdot \\sqrt{b}\\cdot \\sqrt{b}=6b\\\\\\text{Outside}:\\,\\,\\,2\\sqrt{b}\\cdot \\left( -2 \\right)=-4\\sqrt{b}\\\\\\text{Inside}:\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot 3\\sqrt{b}=15\\sqrt{b}\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot \\left( -2 \\right)=-10\\end{array}[\/latex]<\/p>\r\nRecord the terms, and then combine like terms.\r\n<p style=\"text-align: center;\">[latex] 6b-4\\sqrt{b}+15\\sqrt{b}-10[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]6b+11\\sqrt{b}-10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe multiplication works the same way in both problems; you just have to pay attention to the index of the radical (that is, whether the roots are square roots, cube roots, etc.) when multiplying radical expressions.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying Two-Term\u00a0Radical Expressions<\/h3>\r\nTo multiply radical expressions, use the same method as used to multiply polynomials.\r\n<ul>\r\n \t<li>Use the Distributive Property (or, if you prefer, the shortcut FOIL method)<\/li>\r\n \t<li>Remember that [latex] \\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex]<\/li>\r\n \t<li>Combine like terms<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. [latex] \\left( 4{{x}^{2}}+\\sqrt[3]{x} \\right)\\left( \\sqrt[3]{{{x}^{2}}}+2 \\right)[\/latex]\r\n\r\n[reveal-answer q=\"865344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"865344\"]\r\n\r\nUse FOIL to multiply.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4x^{2}\\cdot\\sqrt[3]{x^{2}}=4x^{2}\\sqrt[3]{x^{2}}\\\\\\text{Outside}:\\,\\,\\,4x^{2}\\cdot 2=8x^{2}\\\\\\text{Inside}:\\,\\,\\,\\,\\,\\,\\,\\,\\sqrt[3]{x}\\cdot\\sqrt[3]{x^{2}}=\\sqrt[3]{x^{2}\\cdot x}=\\sqrt[3]{x^{3}}=x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sqrt[3]{x}\\cdot 2=2\\sqrt[3]{x}\\end{array}[\/latex]<\/p>\r\nRecord the terms, and then combine like terms (if possible). Here, there are no like terms to combine.\r\n<p style=\"text-align: center;\">[latex] 4{{x}^{2}}\\sqrt[3]{{{x}^{2}}}+8{{x}^{2}}+x+2\\sqrt[3]{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples of how to multiply two binomials that contain radicals.\r\n\r\nhttps:\/\/youtu.be\/VUWIBk3ga5I\r\n<h2>Summary<\/h2>\r\nTo multiply radical expressions that contain more than one term, use the same method that you use to multiply polynomials. First, use the Distributive Property (or, if you prefer, the shortcut FOIL method) to multiply the terms. Then, apply the rules [latex] \\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex], and [latex] \\sqrt{x}\\cdot \\sqrt{x}=x[\/latex] to multiply and simplify. Finally, combine like terms.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Use the distributive property to multiply multiple term radicals and then simplify<\/li>\n<\/ul>\n<\/div>\n<p>When multiplying multiple term radical expressions, it is important to follow the <strong>Distributive Property of Multiplication<\/strong>, as when you are multiplying regular, non-radical expressions.<\/p>\n<p>Radicals follow the same mathematical rules that other real numbers do. So, although the expression [latex]\\sqrt{x}(3\\sqrt{x}-5)[\/latex] may look different than [latex]a(3a-5)[\/latex], you can treat them the same way.<\/p>\n<p>Let us have a look at how to apply the Distributive Property. First let us do a problem with the variable <i>a<\/i>, and then solve the same problem replacing <i>a<\/i> with [latex]\\sqrt{x}[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]a(3a-5)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q653719\">Show Solution<\/span><\/p>\n<div id=\"q653719\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Distributive Property of Multiplication over Subtraction.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}a(3a)-a(5)\\\\=3a^2-5a\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\sqrt{x}(3\\sqrt{x}-5)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q886472\">Show Solution<\/span><\/p>\n<div id=\"q886472\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Distributive Property of Multiplication over Subtraction.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{x}(3\\sqrt{x})-\\sqrt{x}(5)[\/latex]<\/p>\n<p>Apply the rules of multiplying radicals: [latex]\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex] to multiply [latex]\\sqrt{x}(3\\sqrt{x})[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3\\sqrt{{{x}^{2}}}-5\\sqrt{x}[\/latex]<\/p>\n<p>Be sure to simplify radicals when you can: [latex]\\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex], so [latex]3\\sqrt{{{x}^{2}}}=3\\left| x \\right|[\/latex].<\/p>\n<p>The answer is [latex]3\\left| x \\right|-5\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The answers to the previous two problems should look similar to you. The only difference is that in the second problem, [latex]\\sqrt{x}[\/latex] has replaced the variable <i>a <\/i>(and so [latex]\\left| x \\right|[\/latex] has replaced <i>a<\/i><sup>2<\/sup>). The process of multiplying is very much the same in both problems.<\/p>\n<p>In these next two problems, each term contains a radical.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]7\\sqrt{x}\\left( 2\\sqrt{xy}+\\sqrt{y} \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q732671\">Show Solution<\/span><\/p>\n<div id=\"q732671\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Distributive Property of Multiplication over Addition to multiply each term within parentheses by [latex]7\\sqrt{x}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]7\\sqrt{x}\\left( 2\\sqrt{xy} \\right)+7\\sqrt{x}\\left( \\sqrt{y} \\right)[\/latex]<\/p>\n<p>Apply the rules of multiplying radicals.<\/p>\n<p style=\"text-align: center;\">[latex]7\\cdot 2\\sqrt{{{x}^{2}}y}+7\\sqrt{xy}[\/latex]<\/p>\n<p>[latex]\\sqrt{{{x}^{2}}}=\\left| x \\right|[\/latex], so [latex]\\left| x \\right|[\/latex] can be pulled out of the radical.<\/p>\n<p style=\"text-align: center;\">[latex]14|x|\\sqrt{y}+7\\sqrt{xy}[\/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]{a}\\left( 2\\sqrt[3]{{{a}^{2}}}-4\\sqrt[3]{{{a}^{5}}}+8\\sqrt[3]{{{a}^{8}}} \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q100802\">Show Solution<\/span><\/p>\n<div id=\"q100802\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Distributive Property.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{a}\\left( 2\\sqrt[3]{{{a}^{2}}} \\right)-\\sqrt[3]{a}\\left( 4\\sqrt[3]{{{a}^{5}}} \\right)+\\sqrt[3]{a}\\left( 8\\sqrt[3]{{{a}^{8}}} \\right)[\/latex]<\/p>\n<p>Apply the rules of multiplying radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\sqrt[3]{a\\cdot {{a}^{2}}}-4\\sqrt[3]{a\\cdot {{a}^{5}}}+8\\sqrt[3]{a\\cdot {{a}^{8}}}\\\\2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{a}^{6}}}+8\\sqrt[3]{{{a}^{9}}}\\end{array}[\/latex]<\/p>\n<p>Identify cubes in each of the radicals.<\/p>\n<p style=\"text-align: center;\">[latex]2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{\\left( {{a}^{2}} \\right)}^{3}}}+8\\sqrt[3]{{{\\left( {{a}^{3}} \\right)}^{3}}}[\/latex]<\/p>\n<p>The solution is [latex]2a-4{{a}^{2}}+8{{a}^{3}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of how to multiply radical expressions using the distributive property.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiplying Radical Expressions with Variables  Using Distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hizqmgBjW0k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In all of these examples, multiplication of radicals has been shown following the pattern [latex]\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex]. Then, only after multiplying, some radicals have been simplified\u2014like in the last problem. After you have worked with radical expressions a bit more, you may feel more comfortable identifying quantities such as [latex]\\sqrt{x}\\cdot \\sqrt{x}=x[\/latex] without going through the intermediate step of finding that [latex]\\sqrt{x}\\cdot \\sqrt{x}=\\sqrt{{{x}^{2}}}[\/latex]. In the rest of the examples that follow, though, each step is shown.<\/p>\n<h2>Multiply Binomial Expressions That Contain Radicals<\/h2>\n<p>You can use the same technique for multiplying binomials to multiply binomial\u00a0expressions with radicals.<\/p>\n<p>As a refresher, here is the process for multiplying two binomials. If you like using the expression \u201cFOIL\u201d (First, Outside, Inside, Last) to help you figure out the order in which the terms should be multiplied, you can use it here, too.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. [latex]\\left( 2x+5 \\right)\\left( 3x-2 \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q577475\">Show Solution<\/span><\/p>\n<div id=\"q577475\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Distributive Property.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x\\cdot 3x=6{{x}^{2}}\\\\\\text{Outside}:\\,\\,\\,2x\\cdot \\left( -2 \\right)=-4x\\\\\\text{Inside}:\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot 3x=15x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot \\left( -2 \\right)=-10\\end{array}[\/latex]<\/p>\n<p>Record the terms, and then combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]6{{x}^{2}}-4x+15x-10[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]6{{x}^{2}}+11x-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here is the same problem, with [latex]\\sqrt{b}[\/latex] replacing the variable <i>x<\/i>.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. [latex]\\left( 2\\sqrt{b}+5 \\right)\\left( 3\\sqrt{b}-2 \\right),\\,\\,b\\ge 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q674608\">Show Solution<\/span><\/p>\n<div id=\"q674608\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the Distributive Property to multiply. Simplify using [latex]\\sqrt{x}\\cdot \\sqrt{x}=x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\sqrt{b}\\cdot 3\\sqrt{b}=2\\cdot 3\\cdot \\sqrt{b}\\cdot \\sqrt{b}=6b\\\\\\text{Outside}:\\,\\,\\,2\\sqrt{b}\\cdot \\left( -2 \\right)=-4\\sqrt{b}\\\\\\text{Inside}:\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot 3\\sqrt{b}=15\\sqrt{b}\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\cdot \\left( -2 \\right)=-10\\end{array}[\/latex]<\/p>\n<p>Record the terms, and then combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]6b-4\\sqrt{b}+15\\sqrt{b}-10[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]6b+11\\sqrt{b}-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The multiplication works the same way in both problems; you just have to pay attention to the index of the radical (that is, whether the roots are square roots, cube roots, etc.) when multiplying radical expressions.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying Two-Term\u00a0Radical Expressions<\/h3>\n<p>To multiply radical expressions, use the same method as used to multiply polynomials.<\/p>\n<ul>\n<li>Use the Distributive Property (or, if you prefer, the shortcut FOIL method)<\/li>\n<li>Remember that [latex]\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex]<\/li>\n<li>Combine like terms<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. [latex]\\left( 4{{x}^{2}}+\\sqrt[3]{x} \\right)\\left( \\sqrt[3]{{{x}^{2}}}+2 \\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q865344\">Show Solution<\/span><\/p>\n<div id=\"q865344\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use FOIL to multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4x^{2}\\cdot\\sqrt[3]{x^{2}}=4x^{2}\\sqrt[3]{x^{2}}\\\\\\text{Outside}:\\,\\,\\,4x^{2}\\cdot 2=8x^{2}\\\\\\text{Inside}:\\,\\,\\,\\,\\,\\,\\,\\,\\sqrt[3]{x}\\cdot\\sqrt[3]{x^{2}}=\\sqrt[3]{x^{2}\\cdot x}=\\sqrt[3]{x^{3}}=x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sqrt[3]{x}\\cdot 2=2\\sqrt[3]{x}\\end{array}[\/latex]<\/p>\n<p>Record the terms, and then combine like terms (if possible). Here, there are no like terms to combine.<\/p>\n<p style=\"text-align: center;\">[latex]4{{x}^{2}}\\sqrt[3]{{{x}^{2}}}+8{{x}^{2}}+x+2\\sqrt[3]{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of how to multiply two binomials that contain radicals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Multiplying Binomial Radical Expressions with Variables\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/VUWIBk3ga5I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>To multiply radical expressions that contain more than one term, use the same method that you use to multiply polynomials. First, use the Distributive Property (or, if you prefer, the shortcut FOIL method) to multiply the terms. Then, apply the rules [latex]\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{ab}[\/latex], and [latex]\\sqrt{x}\\cdot \\sqrt{x}=x[\/latex] to multiply and simplify. Finally, combine like terms.<\/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-1657\">\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>Multiplying Radical Expressions with Variables Using Distribution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hizqmgBjW0k\">https:\/\/youtu.be\/hizqmgBjW0k<\/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>Multiplying Binomial Radical Expressions with Variables. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/VUWIBk3ga5I\">https:\/\/youtu.be\/VUWIBk3ga5I<\/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>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>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/li><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><\/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":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Multiplying Radical Expressions with Variables Using 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