Answer
$\dfrac{y^{5}\sqrt[3]{y^2}}{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[3]{\dfrac{y^{17}}{125}}
,$ find a factor of the radicand that is a perfect power of the index. Then extract the root of that factor. Note that all variables are assumed to represent positive real numbers.
$\bf{\text{Solution Details:}}$
Expressing the radicand of the expression above with a factor that is a perfect power of the index and then extracting the root of that factor results to
\begin{array}{l}\require{cancel}
\sqrt[3]{\dfrac{y^{15}}{125}\cdot y^2}
\\\\=
\sqrt[3]{\left( \dfrac{y^{5}}{5} \right)^3\cdot y^2}
\\\\=
\dfrac{y^{5}}{5}\sqrt[3]{y^2}
\\\\=
\dfrac{y^{5}\sqrt[3]{y^2}}{5}
.\end{array}