jmc

The notation $f^{-1}(8)$ stands for a number $x$ such that $f(x)=8$ -- that is, such that $x^3=8$. The only such number is $\sqrt[3]{8}=2$.

The notation $(f(8){)}^{-1}$ stands for $\frac{1}{f(8)}$, which is equal to $\frac{1}{8^3}=\frac{1}{512}$.

Thus, $f^{-1}(8)÷(f(8){)}^{-1}=2÷\frac{1}{512}=2\cdot 512=\boxed{1024}$.