#### Answer

$\dfrac{2x^2\sqrt{2xy}}{y}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To rationalize the denominator of the given expression, $
\sqrt{\dfrac{24x^5}{3y}}
,$ simplify first and then multiply by an expression equal to $1$ which will make the denominator a perfect power of the index.
$\bf{\text{Solution Details:}}$
Simplifying the given radical expression results to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{8x^5}{y}}
.\end{array}
Multiplying the given expression by an expression equal to $1$ which will make the denominator a perfect power of the index and then simplifying the radical result to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{8x^5}{y}\cdot\dfrac{y}{y}}
\\\\=
\sqrt{\dfrac{8x^5y}{y^2}}
.\end{array}
Extracting the perfect root of the index results to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{4x^4}{y^2}\cdot2xy}
\\\\=
\sqrt{\left( \dfrac{2x^2}{y} \right)^2\cdot2xy}
\\\\=
\dfrac{2x^2}{y}\sqrt{2xy}
\\\\=
\dfrac{2x^2\sqrt{2xy}}{y}
.\end{array}