Answer
$\dfrac{|x|\sqrt{10y}}{2y^2}$
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, $
\dfrac{\sqrt{5x^4}}{\sqrt{2x^2y^3}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
=\dfrac{\sqrt{5x^4}}{\sqrt{2x^2y^3}}\cdot\dfrac{\sqrt{2y}}{\sqrt{2y}}
\\\\&=
\dfrac{\sqrt{5x^4(2y)}}{\sqrt{2^2x^2y^4}}
\\\\&=
\dfrac{\sqrt{10x^4y}}{\sqrt{(2xy^2)^2}}
\\\\&=
\dfrac{\sqrt{10x^4y}}{2|x|y^2}
.\end{align*}
Extracting the factors that are perfect powers of the index, the expression above is equivalent to
\begin{align*}\require{cancel}
&
=\dfrac{\sqrt{x^4\cdot10y}}{2|x|y^2}
\\\\&=
\dfrac{\sqrt{(x^2)^2\cdot10y}}{2|x|y^2}
\\\\&=
\dfrac{x^2\sqrt{10y}}{2|x|y^2}
\\\\&=
\dfrac{x^\cancel2\sqrt{10y}}{2\cancel{|x|}y^2}
\\\\&=
\dfrac{|x|\sqrt{10y}}{2y^2}
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{|x|\sqrt{10y}}{2y^2}
$.
