Answer
$\dfrac{x\sqrt{10}}{2y}$
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^4y}}{\sqrt{2x^2y^3}}
,$ is equivalent to
\begin{align*}
&
=\dfrac{\sqrt{5x^4y}}{\sqrt{2x^2y^3}}\cdot\dfrac{\sqrt{2y}}{\sqrt{2y}}
\\\\&=
\dfrac{\sqrt{5x^4y(2y)}}{\sqrt{2^2x^2y^4}}
\\\\&=
\dfrac{\sqrt{10x^4y^2}}{\sqrt{(2xy^2)^2}}
\\\\&=
\dfrac{\sqrt{10x^4y^2}}{2xy^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^4y^2\cdot10}}{2xy^2}
\\\\&=
\dfrac{\sqrt{(x^2y)^2\cdot10}}{2xy^2}
\\\\&=
\dfrac{x^2y\sqrt{10}}{2xy^2}
\\\\&=
\dfrac{x^\cancel2y\sqrt{10}}{2\cancel xy^2}
\\\\&=
\dfrac{xy\sqrt{10}}{2y^2}
\\\\&=
\dfrac{x\cancel y\sqrt{10}}{2y^\cancel2}
\\\\&=
\dfrac{x\sqrt{10}}{2y}
.\end{align*}
Hence, the rationalized-denominator form of the given expression is $
\dfrac{x\sqrt{10}}{2y}
$.
