Answer
$\dfrac{m^3\sqrt[3]{q^2}}{q}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\sqrt[3]{\dfrac{m^9}{q}}
,$ 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{m^9}{q}\cdot\dfrac{q^2}{q^2}}
\\\\=
\sqrt[3]{\dfrac{m^9q^2}{q^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{m^9}{q^3}\cdot q^2}
\\\\=
\sqrt[3]{\left( \dfrac{m^3}{q} \right)^3\cdot q^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{m^3}{q}\sqrt[3]{q^2}
\\\\=
\dfrac{m^3\sqrt[3]{q^2}}{q}
.\end{array}
