Answer
$\dfrac{\sqrt{15y}}{5y}$
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{3xy^2}}{\sqrt{5xy^3}}
,$ is equivalent to
\begin{align*}
&
=\dfrac{\sqrt{3xy^2}}{\sqrt{5xy^3}}\cdot\dfrac{\sqrt{5xy}}{\sqrt{5xy}}
\\\\&=
\dfrac{\sqrt{3xy^2(5xy)}}{\sqrt{5^2x^2y^4}}
\\\\&=
\dfrac{\sqrt{15x^2y^3}}{\sqrt{(5xy^2)^2}}
\\\\&=
\dfrac{\sqrt{15x^2y^3}}{5xy^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^2y^2\cdot15y}}{5xy^2}
\\\\&=
\dfrac{\sqrt{(xy)^2\cdot15y}}{5xy^2}
\\\\&=
\dfrac{xy\sqrt{15y}}{5xy^2}
\\\\&=
\dfrac{\cancel x\cancel y\sqrt{15y}}{5\cancel x\cancel {y^2}y}
\\\\&=
\dfrac{\sqrt{15y}}{5y}
.\end{align*}
Hence, the rationalized-denominator form of the given expression is $
\dfrac{\sqrt{15y}}{5y}
$.
