Answer
$-\frac{4\sqrt[3] a}{b^{3}}$
Work Step by Step
The quotient rule holds that $\sqrt[n] (\frac{a}{b})=\frac{\sqrt[n] a}{\sqrt[n] b}$ (where $\sqrt[n] a$ and $\sqrt[n] b$ are real numbers and $\sqrt[n] b$ is nonzero).
Therefore, $-\sqrt[3] (\frac{64a}{b^{9}})=-\frac{\sqrt[3] (64a)}{\sqrt[3] (b^{9})}=-\frac{\sqrt[3] 64\times\sqrt[3] a}{\sqrt[3] (b^{9})}=-\frac{4\sqrt[3] a}{b^{3}}$
We know that $\sqrt[3] 64=4$, because $4^{3}=64$. Also, we know that $\sqrt[3] (b^{9})=b^{3}$, because $(b^{3})^{3}=b^{3\times3}=b^{9}$.
