Answer
$\dfrac{x\sqrt{y}}{3y^2}$
Work Step by Step
Using the properties of radicals, the given expression, $
\dfrac{\sqrt{5x^2}}{\sqrt{45y^3}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{5x^2}{45y^3}}
\\\\=
\sqrt{\dfrac{\cancel{5}x^2}{\cancel{5}(9)y^3}}
\\\\=
\sqrt{\dfrac{x^2}{9y^3}}
\\\\=
\sqrt{\dfrac{x^2}{9y^2}\cdot\dfrac{1}{y}}
\\\\=
\sqrt{\left(\dfrac{x}{3y}\right)^2\cdot\dfrac{1}{y}}
\\\\=
\dfrac{x}{3y}\sqrt{\dfrac{1}{y}}
\\\\=
\dfrac{x}{3y}\sqrt{\dfrac{1}{y}\cdot\dfrac{y}{y}}
\\\\=
\dfrac{x}{3y}\sqrt{\dfrac{y}{y^2}}
\\\\=
\dfrac{x}{3y}\cdot\dfrac{\sqrt{y}}{\sqrt{y^2}}
\\\\=
\dfrac{x}{3y}\cdot\dfrac{\sqrt{y}}{y}
\\\\=
\dfrac{x\sqrt{y}}{3y^2}
.\end{array}
Note that all variables are assumed to represent positive real numbers.
