Chapter 7 - Section 7.5 - Multiplying and Dividing Radical Expressions - 7.5 Exercises - Page 475: 52

$-\dfrac{5\sqrt{6}}{12}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $\dfrac{-5}{\sqrt{24}} ,$ multiply both the numerator and the denominator by an expression that will make the denominator a perfect power of the index. $\bf{\text{Solution Details:}}$ Expressing the radicand with a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{-5}{\sqrt{4\cdot6}} \\\\ \dfrac{-5}{\sqrt{(2)^2\cdot6}} .\end{array} Multiplying both the numerator and the denominator by an expression that will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{-5}{\sqrt{(2)^2\cdot6}}\cdot\dfrac{\sqrt{6}}{\sqrt{6}} .\end{array} Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{-5\sqrt{6}}{\sqrt{(2)^2\cdot6(6)}} \\\\= \dfrac{-5\sqrt{6}}{\sqrt{(2)^2\cdot(6)^2}} \\\\= \dfrac{-5\sqrt{6}}{2\cdot6} \\\\= \dfrac{-5\sqrt{6}}{12} \\\\= -\dfrac{5\sqrt{6}}{12} .\end{array}

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