{"id":677,"date":"2024-04-24T18:09:51","date_gmt":"2024-04-24T18:09:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=677"},"modified":"2024-11-20T00:53:28","modified_gmt":"2024-11-20T00:53:28","slug":"radicals-and-rational-exponents-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/radicals-and-rational-exponents-learn-it-3\/","title":{"raw":"Radicals and Rational Exponents: Learn It 3","rendered":"Radicals and Rational Exponents: Learn It 3"},"content":{"raw":"

Using the Quotient Rule to Simplify Square Roots<\/h2>\r\nJust as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots.<\/em> It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\\sqrt{\\frac{5}{2}}[\/latex] as [latex]\\frac{\\sqrt{5}}{\\sqrt{2}}[\/latex].\r\n\r\n
\r\n

The Quotient Rule for Simplifying Square Roots<\/h3>\r\nThe square root of the quotient [latex]\\dfrac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex], where [latex]b\\ne 0[\/latex].\r\n
[latex]\\sqrt{\\dfrac{a}{b}}=\\dfrac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/div>\r\n<\/section>
How To: Given a radical expression, use the quotient rule to simplify it.\r\n<\/strong>\r\n
    \r\n \t
  1. Write the radical expression as the quotient of two radical expressions.<\/li>\r\n \t
  2. Simplify the numerator and denominator.<\/li>\r\n<\/ol>\r\n<\/section>
    Simplify the following radical expressions.\r\n
      \r\n \t
    1. [latex]\\sqrt{\\dfrac{5}{36}}[\/latex]<\/li>\r\n \t
    2. [latex]\\sqrt{\\dfrac{2{x}^{2}}{9{y}^{4}}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"786044\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"786044\"]\r\n
        \r\n \t
      1. [latex]\\begin{align*} \\sqrt{\\frac{5}{36}} &= \\frac{\\sqrt{5}}{\\sqrt{36}} & \\text{Separate the square root of the numerator and the denominator.} \\\\ &= \\frac{\\sqrt{5}}{6} & \\text{Simplify the square root of the denominator.} \\end{align*}[\/latex]<\/li>\r\n \t
      2. [latex]\\begin{align*} \\sqrt{\\frac{2x^2}{9y^4}} &= \\frac{\\sqrt{2x^2}}{\\sqrt{9y^4}} & \\text{Separate the square root of the numerator and the denominator.} \\\\ &= \\frac{\\sqrt{2} \\cdot \\sqrt{x^2}}{\\sqrt{9} \\cdot \\sqrt{y^4}} & \\text{Separate the square roots of the factors.} \\\\ &= \\frac{\\sqrt{2} \\cdot |x|}{3 \\cdot y^2} & \\text{Simplify each square root, where } \\sqrt{x^2} = |x| \\text{ and } \\sqrt{y^4} = |y|^2 = y^2. \\\\ &= \\frac{|x|\\sqrt{2}}{3y^2} & \\text{Rearrange the terms for clarity.} \\end{align*}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]18755[\/ohm2_question]<\/section>
        Simplify the expression:\r\n

        [latex]\\dfrac{\\sqrt{234{x}^{11}y}}{\\sqrt{26{x}^{7}y}}[\/latex]<\/p>\r\n

        [reveal-answer q=\"679673\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"679673\"][latex]\\begin{align*} \\frac{\\sqrt{234x^{11}y}}{\\sqrt{26x^7y}} &= \\sqrt{\\frac{234x^{11}y}{26x^7y}} & \\text{Combine the radicals over the fraction.} \\\\ &= \\sqrt{\\frac{234}{26} \\cdot \\frac{x^{11}}{x^7} \\cdot \\frac{y}{y}} & \\text{Separate the terms for simplification.} \\\\ &= \\sqrt{9 \\cdot x^{11-7} \\cdot y^{1-1}} & \\text{Simplify each fraction.} \\\\ &= \\sqrt{9 \\cdot x^4} & \\text{Simplify powers and constants.} \\\\ &= 3x^2 & \\text{Take the square root of 9 and \\(x^4\\).} \\end{align*}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section>

        [ohm2_question hide_question_numbers=1]18756[\/ohm2_question]<\/section>","rendered":"

        Using the Quotient Rule to Simplify Square Roots<\/h2>\n

        Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots.<\/em> It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite [latex]\\sqrt{\\frac{5}{2}}[\/latex] as [latex]\\frac{\\sqrt{5}}{\\sqrt{2}}[\/latex].<\/p>\n

        \n

        The Quotient Rule for Simplifying Square Roots<\/h3>\n

        The square root of the quotient [latex]\\dfrac{a}{b}[\/latex] is equal to the quotient of the square roots of [latex]a[\/latex] and [latex]b[\/latex], where [latex]b\\ne 0[\/latex].<\/p>\n

        [latex]\\sqrt{\\dfrac{a}{b}}=\\dfrac{\\sqrt{a}}{\\sqrt{b}}[\/latex]<\/div>\n<\/section>\n
        How To: Given a radical expression, use the quotient rule to simplify it.
        \n<\/strong><\/p>\n
          \n
        1. Write the radical expression as the quotient of two radical expressions.<\/li>\n
        2. Simplify the numerator and denominator.<\/li>\n<\/ol>\n<\/section>\n
          Simplify the following radical expressions.<\/p>\n
            \n
          1. [latex]\\sqrt{\\dfrac{5}{36}}[\/latex]<\/li>\n
          2. [latex]\\sqrt{\\dfrac{2{x}^{2}}{9{y}^{4}}}[\/latex]<\/li>\n<\/ol>\n