Answer
$\dfrac{x^2\sqrt[3]{y^2}}{y}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\sqrt[3]{\dfrac{x^6}{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[3]{\dfrac{x^6}{y}\cdot\dfrac{y^2}{y^2}}
\\\\=
\sqrt[3]{\dfrac{x^6y^2}{y^3}}
.\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[3]{\dfrac{x^6}{y^3}\cdot y^2}
\\\\=
\sqrt[3]{\left( \dfrac{x^2}{y} \right)^3\cdot y^2}
.\end{array}
Extracting the root of the radicand that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\dfrac{x^2}{y}\sqrt[3]{y^2}
\\\\=
\dfrac{x^2\sqrt[3]{y^2}}{y}
.\end{array}