Answer
$\dfrac{gh^2\sqrt{ghr}}{r^2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt{\dfrac{g^3h^5}{r^3}}
,$ make the denominator a perfect power of the index so that the final result will already be in rationalized form. Then find a factor of the radicand that is a perfect power of the index. Finally, extract the root of that factor.
$\bf{\text{Solution Details:}}$
Multiplying the radicand that will make the denominator a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{g^3h^5}{r^3}\cdot\dfrac{r}{r}}
\\\\=
\sqrt{\dfrac{g^3h^5r}{r^4}}
.\end{array}
Factoring the radicand into an expression that is a perfect power of the index and then extracting its root result to
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{g^2h^4}{r^4}\cdot ghr}
\\\\=
\sqrt{\left( \dfrac{gh^2}{r^2} \right)^2\cdot ghr}
\\\\=
\left| \dfrac{gh^2}{r^2} \right|\sqrt{ghr}
.\end{array}
Since all variables are assumed to be positive, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{gh^2}{r^2}\sqrt{ghr}
\\\\=
\dfrac{gh^2\sqrt{ghr}}{r^2}
.\end{array}
