# Chapter R - Section R.7 - Radical Expressions - R.7 Exercises - Page 68: 61

$\dfrac{h\sqrt[4]{9g^3hr^2}}{3r^2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $\sqrt[4]{\dfrac{g^3h^5}{9r^6}} ,$ make the denominator a perfect power of the index so that the final result will already be in rationalized form. Then find a factor of the radicand that is a perfect power of the index. Finally, extract the root of that factor. $\bf{\text{Solution Details:}}$ Multiplying the radicand that will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt[4]{\dfrac{g^3h^5}{9r^6}\cdot\dfrac{9r^2}{9r^2}} \\\\= \sqrt[4]{\dfrac{9g^3h^5r^2}{81r^8}} .\end{array} Factoring the radicand into an expression that is a perfect power of the index and then extracting its root result to \begin{array}{l}\require{cancel} \sqrt[4]{\dfrac{h^4}{81r^8}\cdot9g^3hr^2} \\\\= \sqrt[4]{\left(\dfrac{h}{3r^2}\right)^4\cdot9g^3hr^2} \\\\= \dfrac{h}{3r^2}\sqrt[4]{9g^3hr^2} \\\\= \dfrac{h\sqrt[4]{9g^3hr^2}}{3r^2} .\end{array} Note that all variables are assumed to have positive values.

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