Answer
$\dfrac{h\sqrt[4]{9g^3hr^2}}{3r^2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[4]{\dfrac{g^3h^5}{9r^6}}
,$ 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[4]{\dfrac{g^3h^5}{9r^6}\cdot\dfrac{9r^2}{9r^2}}
\\\\=
\sqrt[4]{\dfrac{9g^3h^5r^2}{81r^8}}
.\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[4]{\dfrac{h^4}{81r^8}\cdot9g^3hr^2}
\\\\=
\sqrt[4]{\left(\dfrac{h}{3r^2}\right)^4\cdot9g^3hr^2}
\\\\=
\dfrac{h}{3r^2}\sqrt[4]{9g^3hr^2}
\\\\=
\dfrac{h\sqrt[4]{9g^3hr^2}}{3r^2}
.\end{array}
Note that all variables are assumed to have positive values.
