Chapter 7 - Section 7.5 - Multiplying and Dividing Radical Expressions - 7.5 Exercises - Page 475: 56

$-\dfrac{\sqrt{39}}{15}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $-\sqrt{\dfrac{13}{75}} ,$ multiply both the numerator and the denominator by an expression that will make the denominator a perfect power of the index. $\bf{\text{Solution Details:}}$ Converting the radicand as an expression that contains a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} -\sqrt{\dfrac{13}{25\cdot3}} \\\\= -\sqrt{\dfrac{13}{(5)^2\cdot3}} .\end{array} Multiplying the radicand by an expression equal to $1$ which will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} -\sqrt{\dfrac{13}{(5)^2\cdot3}\cdot\dfrac{3}{3}} \\\\= -\sqrt{\dfrac{39}{(5)^2\cdot3^2}} .\end{array} Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to \begin{array}{l}\require{cancel} -\dfrac{\sqrt{39}}{\sqrt{(5)^2\cdot3^2}} \\\\= -\dfrac{\sqrt{39}}{5\cdot3} \\\\= -\dfrac{\sqrt{39}}{15} .\end{array}

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.