Answer
$\dfrac{3\sqrt[4]{y^3}}{y}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
