Intermediate Algebra (12th Edition)

$\dfrac{\sqrt[4]{2yz^3}}{z}$
$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $\sqrt[4]{\dfrac{2y}{z}} ,$ 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[4]{\dfrac{2y}{z}\cdot\dfrac{z^3}{z^3}} \\\\= \sqrt[4]{\dfrac{2yz^3}{z^4}} .\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[4]{\dfrac{1}{z^4}\cdot 2yz^3} \\\\= \sqrt[4]{\left( \dfrac{1}{z}\right)^4\cdot 2yz^3} .\end{array} Extracting the root of the radicand that is a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{1}{z}\sqrt[4]{2yz^3} \\\\= \dfrac{\sqrt[4]{2yz^3}}{z} .\end{array}