Answer
$\dfrac{x\sqrt{xy}}{2y}$
Work Step by Step
Using the properties of radicals, the given expression, $
\dfrac{\sqrt{2x^3}}{\sqrt{8y}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{2x^3}{8y}}
\\\\=
\sqrt{\dfrac{\cancel{2}x^3}{\cancel{2}(4)y}}
\\\\=
\sqrt{\dfrac{x^3}{4y}}
\\\\=
\sqrt{\dfrac{x^2}{4}\cdot\dfrac{x}{y}}
\\\\=
\sqrt{\left(\dfrac{x}{2}\right)^2\cdot\dfrac{x}{y}}
\\\\=
\dfrac{x}{2}\sqrt{\dfrac{x}{y}}
\\\\=
\dfrac{x}{2}\sqrt{\dfrac{x}{y}\cdot\dfrac{y}{y}}
\\\\=
\dfrac{x}{2}\sqrt{\dfrac{xy}{y^2}}
\\\\=
\dfrac{x}{2}\dfrac{\sqrt{xy}}{\sqrt{y^2}}
\\\\=
\dfrac{x}{2}\dfrac{\sqrt{xy}}{\sqrt{(y)^2}}
\\\\=
\dfrac{x}{2}\dfrac{\sqrt{xy}}{y}
\\\\=
\dfrac{x\sqrt{xy}}{2y}
.\end{array}
Note that all variables are assumed to represent positive real numbers.
