{"id":704,"date":"2021-09-07T20:32:27","date_gmt":"2021-09-07T20:32:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=704"},"modified":"2021-11-25T00:20:43","modified_gmt":"2021-11-25T00:20:43","slug":"3-2-5-roots-on-variables","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/3-2-5-roots-on-variables\/","title":{"raw":"3.2.3: Radical Expressions","rendered":"3.2.3: Radical Expressions"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Simplify square roots with variables<\/li>\r\n \t
  • Recognize that by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative<\/li>\r\n<\/ul>\r\n<\/div>\r\n
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    Key words<\/h3>\r\n
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    • Radical expression<\/strong>: an expression that contains radicals<\/li>\r\n<\/ul>\r\n<\/div>\r\nRadical expressions\u00a0<\/strong>are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \\sqrt{16}[\/latex], to more complicated, as in [latex] \\sqrt[3\\;]{250{{x}^{4}}y}[\/latex]. In this section we will discover how to simplify expressions containing square roots.\r\n\r\n\"\"\r\n

      Simplifying Square Roots<\/h2>\r\nWe have already investigated simplifying a radical expression with\u00a0integers by using factoring and the product property of square roots. For example, [latex]\\sqrt{45}=\\sqrt{9\\cdot 5}=\\sqrt{9}\\cdot\\sqrt{5}=3\\sqrt{5}[\/latex]. We can use this same method to simplify a radical term that contains variables. However, we have to be careful.\r\n\r\nConsider the expression [latex] \\sqrt{{{x}^{2}}}[\/latex]. This looks like it should be equal to [latex]x[\/latex], right? Let\u2019s test some values for[latex]x[\/latex]\u00a0<\/i>and see what happens.\r\n\r\nSuppose,\u00a0[latex]x=5[\/latex]. Then\u00a0[latex] \\sqrt{{{x}^{2}}}=\\sqrt{{{5}^{2}}}=\\sqrt{25}=5=x[\/latex]. In this case,\u00a0[latex] \\sqrt{{{x}^{2}}}=x[\/latex].\r\n\r\nNow suppose\u00a0[latex]x=-3[\/latex]. Then\u00a0[latex] \\sqrt{{{x}^{2}}}=\\sqrt{{{(-3)}^{2}}}=\\sqrt{9}=3\\neq x[\/latex]. In this case,\u00a0[latex] \\sqrt{{{x}^{2}}}\\neq x[\/latex].\r\n\r\nThe table shows more examples.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/th>\r\n[latex]x^{2}[\/latex]<\/th>\r\n[latex]\\sqrt{x^{2}}[\/latex]<\/th>\r\n[latex]\\left|x\\right|[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      [latex]\u22125[\/latex]<\/td>\r\n[latex]25[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]\u22122[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]6[\/latex]<\/td>\r\n[latex]36[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]10[\/latex]<\/td>\r\n[latex]100[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that in cases where\u00a0[latex]x[\/latex] is a negative number, [latex]\\sqrt{x^{2}}\\neq{x}[\/latex]! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex].\u00a0We need to consider this fact when simplifying radicals that contain variables, because by definition [latex]\\sqrt{x^{2}}[\/latex]\u00a0is always nonnegative.\r\n\r\nWhen we square any exponential term, we multiply the exponent by 2. For example, [latex]\\left ( y^3\\right )^2=y^{3\\cdot 2}=y^6[\/latex]. This means that in order to take the square root of an exponential term, the exponent must be even.\r\n\r\nLet's consider [latex]\\sqrt{y^6}[\/latex]. We can write\u00a0[latex]y^6[\/latex] as\u00a0[latex]\\left (y^3\\right )^2[\/latex]. Then\u00a0[latex]\\sqrt{y^6}=\\sqrt{\\left (y^3\\right )^2}=\\left | x^3 \\right |[\/latex]. We need the absolute value because the square root must be non-negative.\r\n\r\nNotice that when we square an exponential term, we multiply<\/em> the exponent by [latex]2[\/latex]. Since taking the square root \"undoes\" squaring, we can divide<\/em> the exponent by 2 t take the square root, provided the exponent is even.\r\n
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      the Square Root Of variables<\/h3>\r\n

      [latex]\\sqrt{x^{2}}=\\left|x\\right|[\/latex]<\/p>\r\n

      [latex]\\sqrt{x^{2n}}=\\left | x^n \\right |[\/latex]<\/p>\r\n\r\n<\/div>\r\nNow that we know this, we can simplify a radical expression by using factoring and the product property of square roots.\r\n\r\nThe goal is to find factors under the radical that are perfect squares so that we can simplify.\r\n

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      Example<\/h3>\r\nSimplify. [latex] \\sqrt{9{{x}^{6}}}[\/latex]\r\n

      Solution<\/h4>\r\n
      \r\n\r\nWrite the radicand as perfect square factors. Note how we use the power rule for exponents to write [latex]x^6[\/latex] as a perfect square: [latex]\\left ({x^3}\\right )^2[\/latex]\r\n\r\n[latex] \\sqrt{{{3}^{2}}\\cdot {{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]\r\n\r\nSeparate into individual radicals using the product property.\r\n\r\n[latex] \\sqrt{{{3}^{2}}}\\cdot \\sqrt{{{\\left( {{x}^{3}} \\right)}^{2}}}[\/latex]\r\n\r\nTake the square roots, remembering that [latex] \\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].\r\n\r\n[latex] 3\\left|{{x}^{3}}\\right|[\/latex]\r\n

      Answer<\/h4>\r\n[latex] \\sqrt{9{{x}^{6}}}=3\\left|{{x}^{3}}\\right|[\/latex]\u00a0 \u00a0 \u00a0 We have to keep the absolute value since square roots are never negative.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nLet\u2019s try to simplify another radical expression.\r\n
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      Example<\/h3>\r\nSimplify. [latex] \\sqrt{100{{x}^{2}}{{y}^{4}}}[\/latex]\r\n

      Solution<\/h4>\r\n
      \r\n\r\nFind perfect squares under the radical: exponents that are even.\r\n

      [latex] \\sqrt{10^2\\cdot {x}^{2}\\cdot {y}^{4}}[\/latex]<\/p>\r\nSeparate the perfect squares into individual radicals.\r\n

      [latex] \\sqrt{100}\\cdot \\sqrt{{x}^{2}}\\cdot \\sqrt{({y}^{2})^{2}}[\/latex]<\/p>\r\nSimplify each radical by taking the square root. Remember to put absolute values on variable terms.\r\n

      [latex]10\\cdot\\left|x\\right|\\cdot \\left |{y}^{2}\\right |[\/latex]<\/p>\r\nSimplify. Since [latex]y^2\\geq 0[\/latex], [latex]\\left |{y}^{2}\\right | = y^2[\/latex].\r\n

      [latex]10\\left|x\\right|y^{2}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex] \\sqrt{100{{x}^{2}}{{y}^{4}}}=10\\left| x \\right|{{y}^{2}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nRemember that we can check always check our answer by squaring it to be sure it equals [latex] 100{{x}^{2}}{{y}^{4}}[\/latex].\r\n
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      Example<\/h3>\r\nSimplify. [latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}[\/latex]\r\n
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      Solution<\/h4>\r\nLook for perfect squares:\u00a0 numbers and variables.\r\n\r\n49 is a perfect square since [latex]7^2=49[\/latex]; [latex]x^{10}[\/latex] and [latex]y^8[\/latex] are perfect squares since their exponents are even.\u00a0[latex]x^{10}=\\left (x^5\\right )^2[\/latex] and [latex]y^8=\\left (y^4\\right )^2[\/latex].\r\n\r\n \r\n\r\nSeparate the squared factors into individual radicals.\r\n\r\n[latex] \\sqrt{7^2}\\cdot\\sqrt{({x^5})^2}\\cdot\\sqrt{({y^4})^2}[\/latex]\r\n\r\nTake the square root of each radical using the rule that [latex] \\sqrt{{{x}^{2}}}=\\left|x\\right|[\/latex].\r\n\r\n[latex] 7\\cdot\\left|{{x}^{5}}\\right|\\cdot\\left |{{y}^{4}}\\right |[\/latex]\r\n\r\nSimplifly: [latex]\\left |{{y}^{4}}\\right |=y^4[\/latex], since [latex]y^4\\geq 0[\/latex].\r\n\r\n[latex] 7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]\r\n

      Answer<\/h4>\r\n[latex] \\sqrt{49{{x}^{10}}{{y}^{8}}}=7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\nIn order to check this calculation, we could square [latex]7\\left|{{x}^{5}}\\right|{{y}^{4}}[\/latex], hoping to arrive at [latex] 49{{x}^{10}}{{y}^{8}}[\/latex]. And, in fact, we would get this expression if we evaluated [latex] {\\left({7\\left|{{x}^{5}}\\right|{{y}^{4}}}\\right)^{2}}[\/latex].<\/span>\r\n
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      Try It<\/h3>\r\nSimplify. [latex] \\sqrt{81{{x}^{6}}{{y}^{4}}}[\/latex]\r\n\r\n[reveal-answer q=\"hjm533\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm533\"]\r\n\r\n[latex]9\\left | x^3\\right | y^2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Try It<\/h3>\r\nSimplify. [latex] \\sqrt{144{{x}^{14}}{{y}^{12}}}[\/latex]\r\n\r\n[reveal-answer q=\"hjm083\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm083\"]\r\n\r\n[latex]12\\left | x^7\\right | y^6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSo far we have seen examples that have perfect squares under the radicals: the exponents have all been even. If we have an odd exponent then it is not a perfect square. For example, [latex]x^5[\/latex] is not a perfect square. However, it contains perfect square factors of [latex]x^2[\/latex] and [latex]x^4[\/latex].\r\n

      [latex]x^5=x\\cdot x\\cdot x\\cdot x\\cdot x=x^2\\cdot x^2\\cdot x=x^4\\cdot x[\/latex]<\/p>\r\nEach pair of factors makes a perfect square. This means that we can always write a variable with an odd exponent as the variable to one power less times the variable. For example, [latex]x^7=x^6\\cdot x,\\;y^9=y^8\\cdot y,\\; z^21=z^{20}\\cdot z[\/latex].\r\n

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      perfect square factors of powers of variables<\/h3>\r\n

      [latex]x^n=x^{n-1}\\cdot x[\/latex]<\/p>\r\n\r\n<\/div>\r\nThis means that we can now simplify more radical expressions.\r\n

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      Example<\/h3>\r\nSimplify. [latex] \\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[\/latex]\r\n

      Solution<\/h4>\r\nFactor to find variables with even exponents:\u00a0 [latex] \\sqrt{{{a}^{2}}\\cdot a\\cdot {{b}^{4}}\\cdot{b}\\cdot{{c}^{2}}}[\/latex]\r\n\r\nSeparate the perfect square factors into individual radicals:\u00a0 [latex] \\sqrt{a^2}\\cdot\\sqrt{b^4}\\cdot\\sqrt{c^2}\\cdot \\sqrt{a\\cdot b}[\/latex]\r\n\r\nTake the square root of each radical with a perfect square radicand. Remember that [latex] \\sqrt{{{a}^{2}}}=\\left| a \\right|[\/latex]:\u00a0\u00a0[latex] \\left| a \\right|\\cdot \\left |{b^2}\\right |\\cdot\\left|{c}\\right|\\cdot\\sqrt{a\\cdot b}[\/latex]\r\n\r\nSimplify:\u00a0 [latex] \\left| a\\right |{b^2}\\left | c \\right|\\sqrt{ab}[\/latex]\r\n\r\n \r\n

      Answer<\/h4>\r\n[latex]\\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}} \\left| a\\right |{b^2}\\left | c \\right|\\sqrt{ab}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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      Try It<\/h3>\r\nSimplify. [latex] \\sqrt{{{x}^{4}}{{y}^{9}}{{z}^{3}}}[\/latex]\r\n\r\n[reveal-answer q=\"hjm050\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm050\"]\r\n\r\n[latex]x^2 y^4 \\left |z \\right |\\sqrt{yz}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
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      Try It<\/h3>\r\nSimplify. [latex] \\sqrt{{{m}^{9}}{{n}^{7}}{{p}^{11}}}[\/latex]\r\n\r\n[reveal-answer q=\"hjm516\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm516\"]\r\n\r\n[latex]m^4 \\left | n^3\\right | \\left |p^5 \\right |\\sqrt{mnp}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n ","rendered":"
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      Learning Outcomes<\/h3>\n