{"id":192,"date":"2023-11-08T16:10:20","date_gmt":"2023-11-08T16:10:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/read-or-watch-adding-and-subtracting-radicals\/"},"modified":"2024-08-01T21:05:24","modified_gmt":"2024-08-01T21:05:24","slug":"5-5-radical-expressions-with-multiple-terms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/5-5-radical-expressions-with-multiple-terms\/","title":{"raw":"5.5 Radical Expressions with Multiple Terms","rendered":"5.5 Radical Expressions with Multiple Terms"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract radical expressions.<\/li>\r\n \t<li>Find products of two or more expressions which include radical terms.<\/li>\r\n \t<li>Multiply or divide radical expressions with different indices including expressions written in function notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\nSimilar to the previous two sections, we now ask: Can we add radicals, and if so, how?\u00a0 Consider:\r\n<p style=\"text-align: center;\">[latex]\\require{color}\\sqrt{4}+\\sqrt{9}[\/latex]<\/p>\r\nIf we add the radicands, we get\u00a0[latex]\\color{Red}{\\sqrt{4+9}=\\sqrt{13}}.[\/latex] However, this is <span style=\"color: #ff0000;\">incorrect!<\/span> For this problem we could have instead just evaluated each radical separately and obtained [latex]\\color{Green}{\\sqrt{4}+\\sqrt{9}=2+3=5},[\/latex] which is the <span style=\"color: #339966;\">correct<\/span> answer. <strong>This means adding the radicands is NOT correct<\/strong>. Let's state this here as an important fact:\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"37\" height=\"33\" \/>In general, [latex]\\sqrt[n]{a} + \\sqrt[n]{b} \\neq \\sqrt[n]{a + b}.[\/latex]\r\n\r\n<\/div>\r\nOur exponential properties are not helpful either, because we don't have any properties dealing with multiple terms added or subtracted. In particular,\u00a0[latex](a+b)^n \\neq a^n+b^n[\/latex] as a general rule.\r\n\r\nTo add and subtract radicals, we need to borrow the idea of <strong>like terms<\/strong>. Consider\u00a0[latex]\\sqrt{2} + \\sqrt{2}.[\/latex] We can say that we have 2 copies of\u00a0[latex]\\sqrt{2},[\/latex] or symbolically we have\u00a0[latex]2\\sqrt{2}.[\/latex]\r\n\r\nOne helpful tip is to think of radical expressions like polynomials. In a polynomial, two terms are like terms if the variables and their exponents are the same, for example [latex]3xy^2[\/latex] and [latex]8xy^2[\/latex]. Similarly, two radical expressions are like terms if their indices and radicands are equal, for example\u00a0[latex]3\\sqrt[5]{3x}[\/latex] and\u00a0[latex]-5\\sqrt[5]{3x}.[\/latex]\r\n<div class=\"textbox shaded\">\r\n<h3>LIKE TERMS IN RADICAL EXPRESSIONS<\/h3>\r\nIn an expression with radicals, two radical terms are <strong>like terms<\/strong> if they have the same radicand and the same index.\r\n\r\n<\/div>\r\nHere are some full examples of addition and subtraction.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify each expression by combining like terms.\r\n<ol>\r\n \t<li>[latex] 3\\sqrt{11}+7\\sqrt{11}[\/latex]<\/li>\r\n \t<li>[latex] 5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}[\/latex]<\/li>\r\n \t<li>[latex] 5\\sqrt{13}-3\\sqrt{13}[\/latex]<\/li>\r\n \t<li>[latex] 3\\sqrt{x}+12\\sqrt[3]{x}-7\\sqrt{x}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"971281\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971281\"]\r\n<ol>\r\n \t<li>The two radicals have the same radicand and index. This means you can combine them by adding \"coefficients,\" like in a polynomial.\r\n<p style=\"text-align: center;\">[latex] \\text{3}\\sqrt{11}+ 7\\sqrt{11}=10\\sqrt{11}[\/latex]<\/p>\r\n<\/li>\r\n \t<li>Rearrange terms so that like terms are next to each other. Then add.\r\n<p style=\"text-align: center;\">[latex] 5\\sqrt{2}+2\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}=7\\sqrt{2}+5\\sqrt{3}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We cannot simplify any further because the remaining terms have different radicands.<\/p>\r\n<\/li>\r\n \t<li>The radicands and indices are the same, so these two radical terms can be combined.\r\n<p style=\"text-align: center;\">[latex] 5\\sqrt{13}-3\\sqrt{13}=2\\sqrt{13}[\/latex]<\/p>\r\n<\/li>\r\n \t<li>Rearrange terms so that like terms are next to each other. Then add.\r\n<p style=\"text-align: center;\">[latex] 3\\sqrt{x}-7\\sqrt{x}+12\\sqrt[3]{x}=-4\\sqrt{x}+12\\sqrt[3]{x}[\/latex]<\/p>\r\nThis is the answer since the two remaining terms have different indices.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following videos, we show more examples of how to identify and combine like radical terms.\r\n\r\nhttps:\/\/youtu.be\/ihcZhgm3yBg\r\n\r\nhttps:\/\/youtu.be\/77TR9HsPZ6M\r\n\r\nSometimes it may not be clear immediately that two radical terms can be combined. Consider the following expression.\r\n<p style=\"text-align: center;\">[latex]5\\sqrt{2}+2\\sqrt{18}[\/latex]<\/p>\r\nThey have different radicands, so they cannot be added as written. However, the second radical can be simplified:\r\n<p style=\"text-align: center;\">[latex] \\begin{align} &amp;\\quad 5\\sqrt{2}+2\\sqrt{18}\\\\=&amp;\\quad 5\\sqrt{2}+2\\sqrt{9}\\sqrt{2}\\\\=&amp;\\quad 5\\sqrt{2}+2\\cdot 3\\sqrt{2}\\\\=&amp;\\quad 5\\sqrt{2}+6\\sqrt{2}\\\\=&amp;\\quad 11\\sqrt{2}\\end{align}[\/latex]<\/p>\r\nAfter simplifying the second radical, the expression had two like terms and we could combine them. Always simplify all radicals in the expression first before attempting to find like terms. Here are some more examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify each expression by combining like terms if possible. Assume all variables represent nonnegative quantities.\r\n<ol>\r\n \t<li>[latex] \\sqrt{28}-3\\sqrt{63}[\/latex]<\/li>\r\n \t<li>[latex] 2\\sqrt[3]{40}+\\sqrt[3]{135}[\/latex]<\/li>\r\n \t<li>[latex] x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}[\/latex]<\/li>\r\n \t<li>[latex] 5\\sqrt[4]{{{a}^{5}}b}-a\\sqrt[4]{16ab}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"638886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"638886\"]\r\n<ol>\r\n \t<li style=\"text-align: left;\">First remove all perfect square factors from the radicals, then combine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{28}-3\\sqrt{63}&amp;=\\sqrt{4\\cdot7}-3\\cdot\\sqrt{9\\cdot7}\\\\\r\n&amp;=\\sqrt{2^2}\\cdot\\sqrt{7}-3\\cdot\\sqrt{3^2}\\cdot\\sqrt{7}\\\\\r\n&amp;=2\\sqrt{7}-3\\cdot 3\\cdot\\sqrt{7}\\\\\r\n&amp;=2\\sqrt{7}-9\\sqrt{7}\\\\\r\n&amp;=-7\\sqrt{7}\\end{align}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: left;\">First remove all perfect cubes from the radicals, then combine like terms.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}2\\sqrt[3]{40}+\\sqrt[3]{135}&amp;=2\\sqrt[3]{8\\cdot 5}+\\sqrt[3]{27\\cdot 5}\\\\\r\n&amp;=2\\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{5}+\\sqrt[3]{{{3}^{3}}}\\cdot \\sqrt[3]{5}\\\\\r\n&amp;=2\\cdot 2\\cdot \\sqrt[3]{5}+3\\cdot \\sqrt[3]{5}\\\\\r\n&amp;=4\\sqrt[3]{5}+3\\sqrt[3]{5}\\\\\r\n&amp;=7\\sqrt[3]{5}\\end{align}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: left;\">First remove all perfect cubes from the radicals, then combine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}&amp;=x\\sqrt[3]{x\\cdot {{y}^{3}}\\cdot y}+y\\sqrt[3]{{{x}^{3}}\\cdot x\\cdot y}\\\\&amp;=x\\sqrt[3]{{{y}^{3}}}\\cdot \\sqrt[3]{xy}+y\\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{xy}\\\\&amp;=xy\\cdot \\sqrt[3]{xy}+xy\\cdot \\sqrt[3]{xy}\\\\\r\n&amp;=2xy\\sqrt[3]{xy}\\end{align}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: left;\">Simplify each radical by identifying and pulling out powers of\u00a0[latex]4[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}5\\sqrt[4]{{{a}^{5}}b}-a\\sqrt[4]{16ab}&amp;=5\\sqrt[4]{{{a}^{4}}\\cdot a\\cdot b}-a\\sqrt[4]{{{2}^{4}}\\cdot a\\cdot b}\\\\&amp;=5\\cdot a\\sqrt[4]{a\\cdot b}-a\\cdot 2\\sqrt[4]{a\\cdot b}\\\\&amp;=5a\\sqrt[4]{ab}-2a\\sqrt[4]{ab}\\\\&amp;=3a\\sqrt[4]{ab}\\end{align}[\/latex]<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe following videos show more examples of combining radical expressions that require simplification first.\r\n\r\nhttps:\/\/youtu.be\/S3fGUeALy7E\r\n\r\nhttps:\/\/youtu.be\/6MogonN1PRQ\r\n<h2>Products of Radical Expressions with Multiple Terms<\/h2>\r\nJust like with addition and subtraction, we can use tools from polynomials to work with radical expressions of multiple terms. So, although the expression [latex] \\sqrt{x}(3\\sqrt{x}-5)[\/latex] may look different from [latex] a(3a-5)[\/latex], you can treat them the same way. The second expression can be resolved by distributing,\u00a0[latex] 3a^2-5a[\/latex]. Let's see how to distribute the radical expression.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply [latex] \\sqrt{x}(3\\sqrt{x}-5).[\/latex] Assume all variables represent nonnegative quantities.\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.\r\n<p style=\"text-align: center;\">[latex] \\sqrt{x}(3\\sqrt{x}-5)=\\sqrt{x}\\cdot 3\\sqrt{x}-\\sqrt{x}\\cdot 5[\/latex]<\/p>\r\nNow multiply the radicals and simplify if possible:\r\n<p style=\"text-align: center;\">[latex] \\begin{align}\r\n=&amp;\\quad 3\\sqrt{x^2}-5\\sqrt{x} \\\\\r\n=&amp;\\quad 3x-5\\sqrt{x}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn these next two examples, 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] Assume all variables represent nonnegative quantities.\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 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}+\\sqrt{y} \\right)=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\nPull out the perfect square factor.\r\n<p style=\"text-align: center;\">[latex] =14x\\sqrt{y}+7\\sqrt{xy}[\/latex]<\/p>\r\nThis is the answer because the remaining radicals are not like terms.\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]{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: left;\">[latex] \\sqrt[3]{a}\\left( 2\\sqrt[3]{{{a}^{2}}}-4\\sqrt[3]{{{a}^{5}}}+8\\sqrt[3]{{{a}^{8}}} \\right)=\\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{align}&amp;=2\\sqrt[3]{a\\cdot {{a}^{2}}}-4\\sqrt[3]{a\\cdot {{a}^{5}}}+8\\sqrt[3]{a\\cdot {{a}^{8}}}\\\\&amp;=2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{a}^{6}}}+8\\sqrt[3]{{{a}^{9}}}\\end{align}[\/latex]<\/p>\r\nIdentify cubes in each of the radicals.\r\n<p style=\"text-align: center;\">[latex] \\begin{align}&amp;=2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{\\left( {{a}^{2}} \\right)}^{3}}}+8\\sqrt[3]{{{\\left( {{a}^{3}} \\right)}^{3}}}\\\\&amp;=2a-4{{a}^{2}}+8{{a}^{3}}\\end{align}[\/latex]<\/p>\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\nAfter 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].\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. To multiply\u00a0[latex] \\left( 2x+5 \\right)\\left( 3x-2 \\right),[\/latex] we would typically use the FOIL shortcut. Let's see how that works if we replace each [latex]x[\/latex] with\u00a0[latex]\\sqrt{6}[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply and simplify. [latex] \\left( 2\\sqrt{6}+5 \\right)\\left( 3\\sqrt{6}-2 \\right)[\/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.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;F &amp; &amp;O &amp; &amp;I &amp; &amp;L \\\\ &amp;2\\sqrt{6}\\cdot 3\\sqrt{6} &amp; &amp;2\\sqrt{6}\\cdot \\left( -2 \\right) &amp; &amp;5\\cdot 3\\sqrt{6} &amp; &amp;5\\cdot \\left( -2 \\right) \\\\ &amp;6\\sqrt{6\\cdot 6} &amp; &amp;-4\\sqrt{6} &amp; &amp;15\\sqrt{6} &amp; &amp;-10 \\\\ &amp;36 &amp; &amp;-4\\sqrt{6} &amp; &amp;15\\sqrt{6} &amp; &amp;-10 \\end{align}[\/latex]<\/p>\r\nNow, combine any like terms.\r\n<p style=\"text-align: center;\">[latex] =26 + 11\\sqrt{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nHere are some additional examples which will be extremely useful later.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply and simplify. Assume all variables represent nonnegative quantities.\r\n<ol>\r\n \t<li>[latex]\\left(4\\sqrt{2}-3\\sqrt{7}\\right)\\left(3\\sqrt{2}+\\sqrt{7}\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(5-2\\sqrt{x}\\right)^2[\/latex]<\/li>\r\n \t<li>[latex]\\left(4+\\sqrt{11}\\right)\\left(4-\\sqrt{11}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"865344\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"865344\"]\r\n<ol>\r\n \t<li>FOIL, and then simplify the radicals.[latex]\\begin{align}\\left(4\\sqrt{2}-3\\sqrt{7}\\right)\\left(3\\sqrt{2}+\\sqrt{7}\\right)&amp;=4\\cdot3\\sqrt{2}\\sqrt{2}+4\\sqrt{2}\\sqrt{7}-3\\cdot 3\\sqrt{7}\\sqrt{2}-3\\sqrt{7}\\sqrt{7}\\\\\r\n&amp;=12(2)+4\\sqrt{14}-9\\sqrt{14}-3(7)&amp;&amp;\\color{blue}{\\sqrt{a}\\sqrt{a}=a}\\\\\r\n&amp;=24+4\\sqrt{14}-9\\sqrt{14}-21\\\\\r\n&amp;=3-5\\sqrt{14}&amp;&amp;\\color{blue}{\\textsf{combine like terms}}\\end{align}[\/latex]<\/li>\r\n \t<li>First rewrite the power as binomial multiplication. Then use FOIL and simplify. Alternatively, you could use the formula [latex]\\left(a+b\\right)^2=a^2+2ab+b^2.[\/latex]\r\n[latex]\\begin{align}\\left(5-2\\sqrt{x}\\right)^2&amp;=\\left(5-2\\sqrt{x}\\right)\\left(5-2\\sqrt{x}\\right)\\\\\r\n&amp;=25-5\\cdot 2\\sqrt{x}-5\\cdot 2\\sqrt{x}+4\\sqrt{x}\\sqrt{x}\\\\\r\n&amp;=25-10\\sqrt{x}-10\\sqrt{x}+4x&amp;&amp;\\color{blue}{\\sqrt{x}\\sqrt{x}=x}\\\\\r\n&amp;=25-20\\sqrt{x}+4x&amp;&amp;\\color{blue}{\\textsf{combine like terms}}\\end{align}[\/latex]<\/li>\r\n \t<li>FOIL and then simplify.\r\n[latex]\\begin{align}\\left(4+\\sqrt{11}\\right)\\left(4-\\sqrt{11}\\right)&amp;=16-4\\sqrt{11}+4\\sqrt{11}-\\sqrt{11}\\sqrt{11}\\\\\r\n&amp;=16-4\\sqrt{11}+4\\sqrt{11}-11&amp;&amp;\\color{blue}{\\sqrt{a}\\sqrt{a}=a}\\\\\r\n&amp;=5&amp;&amp;\\color{blue}{\\textsf{combine like terms}}\\end{align}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last part of the previous example, our final answer contained no radicals. The binomials we multiplied are called <strong>conjugates.<\/strong> In the next section we will learn more about conjugates and why it is useful to multiply radical expressions and get whole numbers.\r\n\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>Multiplying and Dividing with Different Indices<\/h2>\r\nIt is possible to multiply and divide radical expressions that have different indices, but we cannot use the same rules we did before. Let's see two examples of how this is possible.\r\n<h3>Example 1: Same Radicand<\/h3>\r\n<p style=\"text-align: center;\">[latex] \\sqrt[3]{5} \\cdot \\sqrt[4]{5} [\/latex]<\/p>\r\nWe use a different exponent property to simplify this: the Product Rule of Exponents, which says [latex]a^m\\cdot a^n=a^{m+n}.[\/latex]\r\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{align} &amp;\\sqrt[3]{5} \\cdot \\sqrt[4]{5}\\\\\r\n=\\quad &amp;5^{\\frac{1}{3}}\\cdot 5^{\\frac{1}{4}}\\\\\r\n=\\quad &amp;5^{\\frac{1}{3}+\\frac{1}{4}}\\\\\r\n=\\quad &amp;5^{\\frac{4}{12}+\\frac{3}{12}}\\\\\r\n=\\quad &amp;5^{\\frac{7}{12}}=\\sqrt[12]{5^7}\\end{align}[\/latex]<\/p>\r\nThis process only works if the radicand is the same in both expressions. We see an important theme which will be used in Example 2 though - often combining radicals turns into a problem of finding a common denominator (which becomes a common index).\r\n<h3>Example 2: Different Radicands<\/h3>\r\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{align}&amp;\\sqrt[3]{5}\\cdot\\sqrt{2}\\\\\r\n=\\quad&amp;5^{\\frac{1}{3}}\\cdot2^{\\frac{1}{2}}\\\\\r\n=\\quad&amp;5^{\\frac{2}{6}}\\cdot2^{\\frac{3}{6}}&amp;&amp;\\color{blue}{\\textsf{Find common denominator}}\\\\\r\n=\\quad&amp;\\left(5^2\\cdot2^3\\right)^{\\frac{1}{6}}&amp;&amp;\\color{blue}{\\textsf{Power of a Product Rule}}\\\\\r\n=\\quad&amp;\\sqrt[6]{200}\r\n\\end{align}[\/latex]<\/p>\r\nThe key here is that the common denominator became the new index on the combined radical. Here are a few more examples to try on your own.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nPerform the indicated operation and simplify. Write answers in radical form. Assume all variables represent positive quantities.\r\n<ol>\r\n \t<li>[latex] \\sqrt[4]{2}\\cdot\\sqrt[5]{4}[\/latex]<\/li>\r\n \t<li>[latex] \\sqrt[7]{ab^4}\\cdot\\sqrt[6]{a^5b}[\/latex]<\/li>\r\n \t<li>Given [latex]f(x)=\\sqrt[3]{x^2}[\/latex] and [latex]g(x)=\\sqrt[6]{x}[\/latex], find and simplify [latex]\\left(\\dfrac{f}{g}\\right)(x).[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"471281\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"471281\"]\r\n<ol>\r\n \t<li>Since we can write [latex]4=2^2[\/latex], this is a situation where the radicands share a common base.\r\n[latex]\\begin{align}\\sqrt[4]{2}\\cdot\\sqrt[5]{4}&amp;=\\sqrt[4]{2}\\cdot\\sqrt[5]{2^2}\\\\\r\n&amp;=2^{1\/4}\\cdot 2^{2\/5}&amp;&amp;\\color{blue}{\\textsf{convert to exponent form}}\\\\\r\n&amp;=2^{\\frac{1}{4}+\\frac{2}{5}}&amp;&amp;\\color{blue}{\\textsf{product rule for exponents}}\\\\\r\n&amp;=2^{\\frac{5}{20}+\\frac{8}{20}}&amp;&amp;\\color{blue}{\\textsf{common denominators}}\\\\\r\n&amp;=2^{13\/20}\\\\\r\n&amp;=\\sqrt[20]{2^{13}}\\end{align}[\/latex]<\/li>\r\n \t<li>The radicands are not equal so we convert to exponent form, get a common denominator, and then factor out the denominator.\r\n[latex]\\begin{align}\\sqrt[7]{ab^4}\\cdot\\sqrt[6]{a^5b}&amp;=\\left(ab^4\\right)^{1\/7}(a^5b)^{1\/6}\\\\\r\n&amp;=\\left(ab^4\\right)^{6\/42}\\left(a^5b\\right)^{7\/42}&amp;&amp;\\color{blue}{\\textsf{common denominators on exponents}}\\\\\r\n&amp;=\\left(\\left(ab^4\\right)^6\\right)^{1\/42}\\left(\\left(a^5b\\right)^7\\right)^{1\/42}&amp;&amp;\\color{blue}{\\textsf{power rule of exponents}}\\\\\r\n&amp;=\\left(\\left(ab^4\\right)^6\\left(a^5b\\right)^7\\right)^{1\/42}&amp;&amp;\\color{blue}{\\textsf{product to a power}}\\\\\r\n&amp;=\\left(a^6b^{24}a^{35}b^7\\right)^{1\/42}&amp;&amp;\\color{blue}{\\textsf{product to a power rule, and power rule}}\\\\\r\n&amp;=\\left(a^{41}b^{31}\\right)^{1\/42}\\\\\r\n&amp;=\\sqrt[42]{a^{41}b^{31}}\\end{align}[\/latex]<\/li>\r\n \t<li>This problem has a common radicand so we convert to exponents and then subtract the exponents.\r\n[latex]\\begin{align}\\left(\\frac{f}{g}\\right)(x)=\\frac{f(x)}{g(x)}&amp;=\\frac{\\sqrt[3]{x^2}}{\\sqrt[6]{x}}\\\\\r\n&amp;=\\frac{x^{2\/3}}{x^{1\/6}}\\\\\r\n&amp;=x^{\\frac{2}{3}-\\frac{1}{6}}&amp;&amp;\\color{blue}{\\textsf{quotient rule of exponents}}\\\\\r\n&amp;=x^{\\frac{4}{6}-\\frac{1}{6}}&amp;&amp;\\color{blue}{\\textsf{common denominators}}\\\\\r\n&amp;=x^{\\frac{3}{6}}\\\\\r\n&amp;=x^{\\frac{1}{2}}\\\\\r\n&amp;=\\sqrt{x}\\end{align}[\/latex]So, [latex]\\left(\\dfrac{f}{g}\\right)(x)=\\sqrt{x}.[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Summary<\/h2>\r\nAdding or subtracting radical terms is possible when the index and the radicand of two or more radicals are the same. Radicals with the same index and radicand are known as like radicals. It is often helpful to treat radicals just as you would treat variables; like radical terms can be added and subtracted in the same way that like variable terms can be added and subtracted. Sometimes, you will need to simplify a radical expression before it is possible to add or subtract like terms.\r\n\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 the shortcut FOIL method) to multiply the terms. Then, apply the rule [latex] \\sqrt[n]{a}\\cdot \\sqrt[n]{b}=\\sqrt[n]{ab}[\/latex] wherever necessary to multiply and simplify. Finally, combine like terms.\r\n\r\nRadicals with different indices can be multiplied or divided, usually by converting the radicals to exponent notation first.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract radical expressions.<\/li>\n<li>Find products of two or more expressions which include radical terms.<\/li>\n<li>Multiply or divide radical expressions with different indices including expressions written in function notation.<\/li>\n<\/ul>\n<\/div>\n<p>Similar to the previous two sections, we now ask: Can we add radicals, and if so, how?\u00a0 Consider:<\/p>\n<p style=\"text-align: center;\">[latex]\\require{color}\\sqrt{4}+\\sqrt{9}[\/latex]<\/p>\n<p>If we add the radicands, we get\u00a0[latex]\\color{Red}{\\sqrt{4+9}=\\sqrt{13}}.[\/latex] However, this is <span style=\"color: #ff0000;\">incorrect!<\/span> For this problem we could have instead just evaluated each radical separately and obtained [latex]\\color{Green}{\\sqrt{4}+\\sqrt{9}=2+3=5},[\/latex] which is the <span style=\"color: #339966;\">correct<\/span> answer. <strong>This means adding the radicands is NOT correct<\/strong>. Let&#8217;s state this here as an important fact:<\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"37\" height=\"33\" \/>In general, [latex]\\sqrt[n]{a} + \\sqrt[n]{b} \\neq \\sqrt[n]{a + b}.[\/latex]<\/p>\n<\/div>\n<p>Our exponential properties are not helpful either, because we don&#8217;t have any properties dealing with multiple terms added or subtracted. In particular,\u00a0[latex](a+b)^n \\neq a^n+b^n[\/latex] as a general rule.<\/p>\n<p>To add and subtract radicals, we need to borrow the idea of <strong>like terms<\/strong>. Consider\u00a0[latex]\\sqrt{2} + \\sqrt{2}.[\/latex] We can say that we have 2 copies of\u00a0[latex]\\sqrt{2},[\/latex] or symbolically we have\u00a0[latex]2\\sqrt{2}.[\/latex]<\/p>\n<p>One helpful tip is to think of radical expressions like polynomials. In a polynomial, two terms are like terms if the variables and their exponents are the same, for example [latex]3xy^2[\/latex] and [latex]8xy^2[\/latex]. Similarly, two radical expressions are like terms if their indices and radicands are equal, for example\u00a0[latex]3\\sqrt[5]{3x}[\/latex] and\u00a0[latex]-5\\sqrt[5]{3x}.[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>LIKE TERMS IN RADICAL EXPRESSIONS<\/h3>\n<p>In an expression with radicals, two radical terms are <strong>like terms<\/strong> if they have the same radicand and the same index.<\/p>\n<\/div>\n<p>Here are some full examples of addition and subtraction.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify each expression by combining like terms.<\/p>\n<ol>\n<li>[latex]3\\sqrt{11}+7\\sqrt{11}[\/latex]<\/li>\n<li>[latex]5\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}+2\\sqrt{2}[\/latex]<\/li>\n<li>[latex]5\\sqrt{13}-3\\sqrt{13}[\/latex]<\/li>\n<li>[latex]3\\sqrt{x}+12\\sqrt[3]{x}-7\\sqrt{x}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q971281\">Show Solution<\/span><\/p>\n<div id=\"q971281\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The two radicals have the same radicand and index. This means you can combine them by adding &#8220;coefficients,&#8221; like in a polynomial.\n<p style=\"text-align: center;\">[latex]\\text{3}\\sqrt{11}+ 7\\sqrt{11}=10\\sqrt{11}[\/latex]<\/p>\n<\/li>\n<li>Rearrange terms so that like terms are next to each other. Then add.\n<p style=\"text-align: center;\">[latex]5\\sqrt{2}+2\\sqrt{2}+\\sqrt{3}+4\\sqrt{3}=7\\sqrt{2}+5\\sqrt{3}[\/latex]<\/p>\n<p style=\"text-align: left;\">We cannot simplify any further because the remaining terms have different radicands.<\/p>\n<\/li>\n<li>The radicands and indices are the same, so these two radical terms can be combined.\n<p style=\"text-align: center;\">[latex]5\\sqrt{13}-3\\sqrt{13}=2\\sqrt{13}[\/latex]<\/p>\n<\/li>\n<li>Rearrange terms so that like terms are next to each other. Then add.\n<p style=\"text-align: center;\">[latex]3\\sqrt{x}-7\\sqrt{x}+12\\sqrt[3]{x}=-4\\sqrt{x}+12\\sqrt[3]{x}[\/latex]<\/p>\n<p>This is the answer since the two remaining terms have different indices.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following videos, we show more examples of how to identify and combine like radical terms.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Adding Radicals (Basic With No Simplifying)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ihcZhgm3yBg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Subtracting Radicals (Basic With No Simplifying)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/77TR9HsPZ6M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Sometimes it may not be clear immediately that two radical terms can be combined. Consider the following expression.<\/p>\n<p style=\"text-align: center;\">[latex]5\\sqrt{2}+2\\sqrt{18}[\/latex]<\/p>\n<p>They have different radicands, so they cannot be added as written. However, the second radical can be simplified:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &\\quad 5\\sqrt{2}+2\\sqrt{18}\\\\=&\\quad 5\\sqrt{2}+2\\sqrt{9}\\sqrt{2}\\\\=&\\quad 5\\sqrt{2}+2\\cdot 3\\sqrt{2}\\\\=&\\quad 5\\sqrt{2}+6\\sqrt{2}\\\\=&\\quad 11\\sqrt{2}\\end{align}[\/latex]<\/p>\n<p>After simplifying the second radical, the expression had two like terms and we could combine them. Always simplify all radicals in the expression first before attempting to find like terms. Here are some more examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify each expression by combining like terms if possible. Assume all variables represent nonnegative quantities.<\/p>\n<ol>\n<li>[latex]\\sqrt{28}-3\\sqrt{63}[\/latex]<\/li>\n<li>[latex]2\\sqrt[3]{40}+\\sqrt[3]{135}[\/latex]<\/li>\n<li>[latex]x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}[\/latex]<\/li>\n<li>[latex]5\\sqrt[4]{{{a}^{5}}b}-a\\sqrt[4]{16ab}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q638886\">Show Solution<\/span><\/p>\n<div id=\"q638886\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"text-align: left;\">First remove all perfect square factors from the radicals, then combine like terms.\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{28}-3\\sqrt{63}&=\\sqrt{4\\cdot7}-3\\cdot\\sqrt{9\\cdot7}\\\\  &=\\sqrt{2^2}\\cdot\\sqrt{7}-3\\cdot\\sqrt{3^2}\\cdot\\sqrt{7}\\\\  &=2\\sqrt{7}-3\\cdot 3\\cdot\\sqrt{7}\\\\  &=2\\sqrt{7}-9\\sqrt{7}\\\\  &=-7\\sqrt{7}\\end{align}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: left;\">First remove all perfect cubes from the radicals, then combine like terms.\n<p style=\"text-align: center;\">[latex]\\begin{align}2\\sqrt[3]{40}+\\sqrt[3]{135}&=2\\sqrt[3]{8\\cdot 5}+\\sqrt[3]{27\\cdot 5}\\\\  &=2\\sqrt[3]{{{2}^{3}}}\\cdot \\sqrt[3]{5}+\\sqrt[3]{{{3}^{3}}}\\cdot \\sqrt[3]{5}\\\\  &=2\\cdot 2\\cdot \\sqrt[3]{5}+3\\cdot \\sqrt[3]{5}\\\\  &=4\\sqrt[3]{5}+3\\sqrt[3]{5}\\\\  &=7\\sqrt[3]{5}\\end{align}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: left;\">First remove all perfect cubes from the radicals, then combine like terms.\n<p style=\"text-align: center;\">[latex]\\begin{align}x\\sqrt[3]{x{{y}^{4}}}+y\\sqrt[3]{{{x}^{4}}y}&=x\\sqrt[3]{x\\cdot {{y}^{3}}\\cdot y}+y\\sqrt[3]{{{x}^{3}}\\cdot x\\cdot y}\\\\&=x\\sqrt[3]{{{y}^{3}}}\\cdot \\sqrt[3]{xy}+y\\sqrt[3]{{{x}^{3}}}\\cdot \\sqrt[3]{xy}\\\\&=xy\\cdot \\sqrt[3]{xy}+xy\\cdot \\sqrt[3]{xy}\\\\  &=2xy\\sqrt[3]{xy}\\end{align}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: left;\">Simplify each radical by identifying and pulling out powers of\u00a0[latex]4[\/latex].\n<p style=\"text-align: center;\">[latex]\\begin{align}5\\sqrt[4]{{{a}^{5}}b}-a\\sqrt[4]{16ab}&=5\\sqrt[4]{{{a}^{4}}\\cdot a\\cdot b}-a\\sqrt[4]{{{2}^{4}}\\cdot a\\cdot b}\\\\&=5\\cdot a\\sqrt[4]{a\\cdot b}-a\\cdot 2\\sqrt[4]{a\\cdot b}\\\\&=5a\\sqrt[4]{ab}-2a\\sqrt[4]{ab}\\\\&=3a\\sqrt[4]{ab}\\end{align}[\/latex]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following videos show more examples of combining radical expressions that require simplification first.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Adding Radicals That Requires Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/S3fGUeALy7E?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Subtracting Radicals That Requires Simplifying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/6MogonN1PRQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Products of Radical Expressions with Multiple Terms<\/h2>\n<p>Just like with addition and subtraction, we can use tools from polynomials to work with radical expressions of multiple terms. So, although the expression [latex]\\sqrt{x}(3\\sqrt{x}-5)[\/latex] may look different from [latex]a(3a-5)[\/latex], you can treat them the same way. The second expression can be resolved by distributing,\u00a0[latex]3a^2-5a[\/latex]. Let&#8217;s see how to distribute the radical expression.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply [latex]\\sqrt{x}(3\\sqrt{x}-5).[\/latex] Assume all variables represent nonnegative quantities.<\/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.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt{x}(3\\sqrt{x}-5)=\\sqrt{x}\\cdot 3\\sqrt{x}-\\sqrt{x}\\cdot 5[\/latex]<\/p>\n<p>Now multiply the radicals and simplify if possible:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}  =&\\quad 3\\sqrt{x^2}-5\\sqrt{x} \\\\  =&\\quad 3x-5\\sqrt{x}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In these next two examples, 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] Assume all variables represent nonnegative quantities.<\/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 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}+\\sqrt{y} \\right)=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>Pull out the perfect square factor.<\/p>\n<p style=\"text-align: center;\">[latex]=14x\\sqrt{y}+7\\sqrt{xy}[\/latex]<\/p>\n<p>This is the answer because the remaining radicals are not like terms.<\/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: left;\">[latex]\\sqrt[3]{a}\\left( 2\\sqrt[3]{{{a}^{2}}}-4\\sqrt[3]{{{a}^{5}}}+8\\sqrt[3]{{{a}^{8}}} \\right)=\\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{align}&=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{align}[\/latex]<\/p>\n<p>Identify cubes in each of the radicals.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&=2\\sqrt[3]{{{a}^{3}}}-4\\sqrt[3]{{{\\left( {{a}^{2}} \\right)}^{3}}}+8\\sqrt[3]{{{\\left( {{a}^{3}} \\right)}^{3}}}\\\\&=2a-4{{a}^{2}}+8{{a}^{3}}\\end{align}[\/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-5\" 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>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].<\/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. To multiply\u00a0[latex]\\left( 2x+5 \\right)\\left( 3x-2 \\right),[\/latex] we would typically use the FOIL shortcut. Let&#8217;s see how that works if we replace each [latex]x[\/latex] with\u00a0[latex]\\sqrt{6}[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply and simplify. [latex]\\left( 2\\sqrt{6}+5 \\right)\\left( 3\\sqrt{6}-2 \\right)[\/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.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&F & &O & &I & &L \\\\ &2\\sqrt{6}\\cdot 3\\sqrt{6} & &2\\sqrt{6}\\cdot \\left( -2 \\right) & &5\\cdot 3\\sqrt{6} & &5\\cdot \\left( -2 \\right) \\\\ &6\\sqrt{6\\cdot 6} & &-4\\sqrt{6} & &15\\sqrt{6} & &-10 \\\\ &36 & &-4\\sqrt{6} & &15\\sqrt{6} & &-10 \\end{align}[\/latex]<\/p>\n<p>Now, combine any like terms.<\/p>\n<p style=\"text-align: center;\">[latex]=26 + 11\\sqrt{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Here are some additional examples which will be extremely useful later.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply and simplify. Assume all variables represent nonnegative quantities.<\/p>\n<ol>\n<li>[latex]\\left(4\\sqrt{2}-3\\sqrt{7}\\right)\\left(3\\sqrt{2}+\\sqrt{7}\\right)[\/latex]<\/li>\n<li>[latex]\\left(5-2\\sqrt{x}\\right)^2[\/latex]<\/li>\n<li>[latex]\\left(4+\\sqrt{11}\\right)\\left(4-\\sqrt{11}\\right)[\/latex]<\/li>\n<\/ol>\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<ol>\n<li>FOIL, and then simplify the radicals.[latex]\\begin{align}\\left(4\\sqrt{2}-3\\sqrt{7}\\right)\\left(3\\sqrt{2}+\\sqrt{7}\\right)&=4\\cdot3\\sqrt{2}\\sqrt{2}+4\\sqrt{2}\\sqrt{7}-3\\cdot 3\\sqrt{7}\\sqrt{2}-3\\sqrt{7}\\sqrt{7}\\\\  &=12(2)+4\\sqrt{14}-9\\sqrt{14}-3(7)&&\\color{blue}{\\sqrt{a}\\sqrt{a}=a}\\\\  &=24+4\\sqrt{14}-9\\sqrt{14}-21\\\\  &=3-5\\sqrt{14}&&\\color{blue}{\\textsf{combine like terms}}\\end{align}[\/latex]<\/li>\n<li>First rewrite the power as binomial multiplication. Then use FOIL and simplify. Alternatively, you could use the formula [latex]\\left(a+b\\right)^2=a^2+2ab+b^2.[\/latex]<br \/>\n[latex]\\begin{align}\\left(5-2\\sqrt{x}\\right)^2&=\\left(5-2\\sqrt{x}\\right)\\left(5-2\\sqrt{x}\\right)\\\\  &=25-5\\cdot 2\\sqrt{x}-5\\cdot 2\\sqrt{x}+4\\sqrt{x}\\sqrt{x}\\\\  &=25-10\\sqrt{x}-10\\sqrt{x}+4x&&\\color{blue}{\\sqrt{x}\\sqrt{x}=x}\\\\  &=25-20\\sqrt{x}+4x&&\\color{blue}{\\textsf{combine like terms}}\\end{align}[\/latex]<\/li>\n<li>FOIL and then simplify.<br \/>\n[latex]\\begin{align}\\left(4+\\sqrt{11}\\right)\\left(4-\\sqrt{11}\\right)&=16-4\\sqrt{11}+4\\sqrt{11}-\\sqrt{11}\\sqrt{11}\\\\  &=16-4\\sqrt{11}+4\\sqrt{11}-11&&\\color{blue}{\\sqrt{a}\\sqrt{a}=a}\\\\  &=5&&\\color{blue}{\\textsf{combine like terms}}\\end{align}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last part of the previous example, our final answer contained no radicals. The binomials we multiplied are called <strong>conjugates.<\/strong> In the next section we will learn more about conjugates and why it is useful to multiply radical expressions and get whole numbers.<\/p>\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-6\" 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>Multiplying and Dividing with Different Indices<\/h2>\n<p>It is possible to multiply and divide radical expressions that have different indices, but we cannot use the same rules we did before. Let&#8217;s see two examples of how this is possible.<\/p>\n<h3>Example 1: Same Radicand<\/h3>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{5} \\cdot \\sqrt[4]{5}[\/latex]<\/p>\n<p>We use a different exponent property to simplify this: the Product Rule of Exponents, which says [latex]a^m\\cdot a^n=a^{m+n}.[\/latex]<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{align} &\\sqrt[3]{5} \\cdot \\sqrt[4]{5}\\\\  =\\quad &5^{\\frac{1}{3}}\\cdot 5^{\\frac{1}{4}}\\\\  =\\quad &5^{\\frac{1}{3}+\\frac{1}{4}}\\\\  =\\quad &5^{\\frac{4}{12}+\\frac{3}{12}}\\\\  =\\quad &5^{\\frac{7}{12}}=\\sqrt[12]{5^7}\\end{align}[\/latex]<\/p>\n<p>This process only works if the radicand is the same in both expressions. We see an important theme which will be used in Example 2 though &#8211; often combining radicals turns into a problem of finding a common denominator (which becomes a common index).<\/p>\n<h3>Example 2: Different Radicands<\/h3>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{align}&\\sqrt[3]{5}\\cdot\\sqrt{2}\\\\  =\\quad&5^{\\frac{1}{3}}\\cdot2^{\\frac{1}{2}}\\\\  =\\quad&5^{\\frac{2}{6}}\\cdot2^{\\frac{3}{6}}&&\\color{blue}{\\textsf{Find common denominator}}\\\\  =\\quad&\\left(5^2\\cdot2^3\\right)^{\\frac{1}{6}}&&\\color{blue}{\\textsf{Power of a Product Rule}}\\\\  =\\quad&\\sqrt[6]{200}  \\end{align}[\/latex]<\/p>\n<p>The key here is that the common denominator became the new index on the combined radical. Here are a few more examples to try on your own.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Perform the indicated operation and simplify. Write answers in radical form. Assume all variables represent positive quantities.<\/p>\n<ol>\n<li>[latex]\\sqrt[4]{2}\\cdot\\sqrt[5]{4}[\/latex]<\/li>\n<li>[latex]\\sqrt[7]{ab^4}\\cdot\\sqrt[6]{a^5b}[\/latex]<\/li>\n<li>Given [latex]f(x)=\\sqrt[3]{x^2}[\/latex] and [latex]g(x)=\\sqrt[6]{x}[\/latex], find and simplify [latex]\\left(\\dfrac{f}{g}\\right)(x).[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q471281\">Show Solution<\/span><\/p>\n<div id=\"q471281\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Since we can write [latex]4=2^2[\/latex], this is a situation where the radicands share a common base.<br \/>\n[latex]\\begin{align}\\sqrt[4]{2}\\cdot\\sqrt[5]{4}&=\\sqrt[4]{2}\\cdot\\sqrt[5]{2^2}\\\\  &=2^{1\/4}\\cdot 2^{2\/5}&&\\color{blue}{\\textsf{convert to exponent form}}\\\\  &=2^{\\frac{1}{4}+\\frac{2}{5}}&&\\color{blue}{\\textsf{product rule for exponents}}\\\\  &=2^{\\frac{5}{20}+\\frac{8}{20}}&&\\color{blue}{\\textsf{common denominators}}\\\\  &=2^{13\/20}\\\\  &=\\sqrt[20]{2^{13}}\\end{align}[\/latex]<\/li>\n<li>The radicands are not equal so we convert to exponent form, get a common denominator, and then factor out the denominator.<br \/>\n[latex]\\begin{align}\\sqrt[7]{ab^4}\\cdot\\sqrt[6]{a^5b}&=\\left(ab^4\\right)^{1\/7}(a^5b)^{1\/6}\\\\  &=\\left(ab^4\\right)^{6\/42}\\left(a^5b\\right)^{7\/42}&&\\color{blue}{\\textsf{common denominators on exponents}}\\\\  &=\\left(\\left(ab^4\\right)^6\\right)^{1\/42}\\left(\\left(a^5b\\right)^7\\right)^{1\/42}&&\\color{blue}{\\textsf{power rule of exponents}}\\\\  &=\\left(\\left(ab^4\\right)^6\\left(a^5b\\right)^7\\right)^{1\/42}&&\\color{blue}{\\textsf{product to a power}}\\\\  &=\\left(a^6b^{24}a^{35}b^7\\right)^{1\/42}&&\\color{blue}{\\textsf{product to a power rule, and power rule}}\\\\  &=\\left(a^{41}b^{31}\\right)^{1\/42}\\\\  &=\\sqrt[42]{a^{41}b^{31}}\\end{align}[\/latex]<\/li>\n<li>This problem has a common radicand so we convert to exponents and then subtract the exponents.<br \/>\n[latex]\\begin{align}\\left(\\frac{f}{g}\\right)(x)=\\frac{f(x)}{g(x)}&=\\frac{\\sqrt[3]{x^2}}{\\sqrt[6]{x}}\\\\  &=\\frac{x^{2\/3}}{x^{1\/6}}\\\\  &=x^{\\frac{2}{3}-\\frac{1}{6}}&&\\color{blue}{\\textsf{quotient rule of exponents}}\\\\  &=x^{\\frac{4}{6}-\\frac{1}{6}}&&\\color{blue}{\\textsf{common denominators}}\\\\  &=x^{\\frac{3}{6}}\\\\  &=x^{\\frac{1}{2}}\\\\  &=\\sqrt{x}\\end{align}[\/latex]So, [latex]\\left(\\dfrac{f}{g}\\right)(x)=\\sqrt{x}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>Adding or subtracting radical terms is possible when the index and the radicand of two or more radicals are the same. Radicals with the same index and radicand are known as like radicals. It is often helpful to treat radicals just as you would treat variables; like radical terms can be added and subtracted in the same way that like variable terms can be added and subtracted. Sometimes, you will need to simplify a radical expression before it is possible to add or subtract like terms.<\/p>\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 the shortcut FOIL method) to multiply the terms. Then, apply the rule [latex]\\sqrt[n]{a}\\cdot \\sqrt[n]{b}=\\sqrt[n]{ab}[\/latex] wherever necessary to multiply and simplify. Finally, combine like terms.<\/p>\n<p>Radicals with different indices can be multiplied or divided, usually by converting the radicals to exponent notation first.<\/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-192\">\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>Adding Radicals (Basic With No Simplifying). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ihcZhgm3yBg\">https:\/\/youtu.be\/ihcZhgm3yBg<\/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>Adding Radicals That Requires Simplifying. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/S3fGUeALy7E\">https:\/\/youtu.be\/S3fGUeALy7E<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Subtracting Radicals (Basic With No Simplifying). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/77TR9HsPZ6M\">https:\/\/youtu.be\/77TR9HsPZ6M<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Subtracting Radicals That Requires Simplifying. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/6MogonN1PRQ\">https:\/\/youtu.be\/6MogonN1PRQ<\/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":395986,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Adding Radicals (Basic With No Simplifying)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ihcZhgm3yBg\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Adding Radicals That Requires Simplifying\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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