Answer
$$30y$$
Work Step by Step
RECALL:
(1) For $a\gt 0, b \gt0$, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.
(2) If $n$ is odd, $\sqrt[n]{a^n} = a$.
(3) If $n$ is even, $\sqrt[n]{a^n} = |a|$.
Using rule (1) above gives:
\begin{align*}
5\sqrt[3]{9y^2} \cdot \sqrt[3]{24y}&=5\sqrt[3]{(9y^2)(24y)}\\
&=5\sqrt[3]{216y^3}\\
&=5\sqrt[3]{(6y)(6y)(6y)}\\
&=5\sqrt[3]{\left(6y\right)^3}\\
\end{align*}
Use rule (2) above to obtain:
\begin{align*}
5\sqrt[3]{\left(6y\right)^3}&=30y
\end{align*}