Answer
$\dfrac{4x^3}{y\sqrt{2x}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the numerator of the given expression, $
\sqrt{\dfrac{24x^5}{3y^2}}
,$ simplify first by extracting the factor that is a perfect power of the index. Then multiply by an expression equal to $1$ which will make the numerator a perfect power of the index. Then extract again the root of the factor that is a perfect power of the index.
$\bf{\text{Solution Details:}}$
Simplifying the radicand of the given radical expression results to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{8x^5}{y^2}}
\\\\=
\sqrt{\dfrac{4x^4}{y^2}\cdot2x}
\\\\=
\sqrt{\left(\dfrac{2x^2}{y}\right)^2\cdot2x}
\\\\=
\dfrac{2x^2}{y}\sqrt{2x}
\\\\=
\dfrac{2x^2\sqrt{2x}}{y}
.\end{array}
Multiplying the given expression by an expression equal to $1$ which will make the numerator a perfect power of the index results to
\begin{array}{l}\require{cancel}
\dfrac{2x^2\sqrt{2x}}{y}\cdot\dfrac{\sqrt{2x}}{\sqrt{2x}}
\\\\=
\dfrac{2x^2\sqrt{4x^2}}{y\sqrt{2x}}
.\end{array}
Extracting the root of the radicand that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\dfrac{2x^2\sqrt{(2x)^2}}{y\sqrt{2x}}
\\\\=
\dfrac{2x^2(2x)}{y\sqrt{2x}}
\\\\=
\dfrac{4x^3}{y\sqrt{2x}}
.\end{array}