Answer
$\dfrac{\sqrt[3]{45x^2}}{3x}$
Work Step by Step
Multiplying the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index, the given expression, $
\sqrt[3]{\dfrac{5}{3x}}
,$ is equivalent to
\begin{align*}
&
=\sqrt[3]{\dfrac{5}{3x}\cdot\dfrac{3^2x^2}{3^2x^2}}
\\\\&=
\sqrt[3]{\dfrac{45x^2}{3^3x^3}}
\\\\&=
\sqrt[3]{\dfrac{45x^2}{(3x)^3}}
.\end{align*}
Using $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}},$ the expression above is equivalent to
\begin{align*}
&
=\dfrac{\sqrt[3]{45x^2}}{\sqrt[3]{(3x)^3}}
\\\\&=
\dfrac{\sqrt[3]{45x^2}}{3x}
.\end{align*}
Hence, the rationalized-denominator form of the given expression is $
\dfrac{\sqrt[3]{45x^2}}{3x}
$.