## Intermediate Algebra (12th Edition)

$\dfrac{3\sqrt[4]{y^3}}{y}$
$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $\sqrt[4]{\dfrac{81}{y}} ,$ 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{81}{y}\cdot\dfrac{y^3}{y^3}} \\\\= \sqrt[4]{\dfrac{81y^3}{y^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{81}{y^4}\cdot y^3} \\\\= \sqrt[4]{\left( \dfrac{3}{y} \right)^4\cdot y^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{3}{y}\sqrt[4]{y^3} \\\\= \dfrac{3\sqrt[4]{y^3}}{y} .\end{array}