Answer
$\dfrac{\sqrt[3]{12}}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\sqrt[3]{\dfrac{4}{9}}
,$ 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{4}{9}\cdot\dfrac{3}{3}}
\\\\=
\sqrt[3]{\dfrac{12}{27}}
\\\\=
\sqrt[3]{\dfrac{12}{(3)^3}}
.\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]{12}}{\sqrt[3]{(3)^3}}
\\\\=
\dfrac{\sqrt[3]{12}}{3}
.\end{array}