Answer
$\dfrac{\sqrt[3]{18}}{4}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\sqrt[3]{\dfrac{9}{32}}
,$ 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{9}{2^5}}
\\\\=
\sqrt[3]{\dfrac{9}{2^5}\cdot\dfrac{2}{2}}
\\\\=
\sqrt[3]{\dfrac{18}{2^6}}
.\end{array}
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[3]{18}}{\sqrt[3]{2^6}}
\\\\=
\dfrac{\sqrt[3]{18}}{\sqrt[3]{(2^2)^3}}
\\\\=
\dfrac{\sqrt[3]{18}}{2^2}
\\\\=
\dfrac{\sqrt[3]{18}}{4}
.\end{array}
