# Chapter 7 - Section 7.5 - Multiplying and Dividing Radical Expressions - 7.5 Exercises: 68

$-\dfrac{5m\sqrt{3mp}}{p}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $-\sqrt{\dfrac{75m^3}{p}} ,$ multiply the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index. $\bf{\text{Solution Details:}}$ Multiplying the numerator and the denominator 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{75m^3}{p}\cdot\dfrac{p}{p}} \\\\= -\sqrt{\dfrac{75m^3p}{p^2}} .\end{array} Rewriting the radicand using an expression that contains a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} -\sqrt{\dfrac{25m^2}{p^2}\cdot3mp} \\\\= -\sqrt{\left(\dfrac{5m}{p}\right)^2\cdot3mp} .\end{array} Extracting the root of the radicand that is a perfect power of the index results to \begin{array}{l}\require{cancel} -\dfrac{5m}{p}\sqrt{3mp} \\\\= -\dfrac{5m\sqrt{3mp}}{p} .\end{array}

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