Simplification algebraic of a cube root. I am trying to simplify this:
$$\frac{1000}{\pi \cdot (\frac{500}{\pi})^{\frac{2}{3}}}$$
and I think it becomes:
$$2 \cdot \sqrt[3]{\frac{500}{\pi}}$$
I basically thought we cube root the $\frac{500}{\pi}$ and then multiply the denominator by $\frac{500}{\pi}$ which could cancel out some stuff. This is how I thought about it:
$$\frac{1000}{\pi \cdot \sqrt[3]{\frac{500}{\pi}} \cdot \frac{500}{\pi}}$$
$$ = \frac{1000 \cdot \pi}{\pi \cdot \sqrt[3]{\frac{500}{\pi}} \cdot 500}$$
Is there a better way to do this cancellation?

$$\frac{1000}{ \pi \cdot (\frac{500}{\pi})^{\frac{2}{3}} }
=\frac{1000}{\pi^{\frac{1}{3}}\cdot 2^{-\frac{2}{3}}\cdot 2^{\frac{2}{3}}\cdot 500^{\frac{2}{3}}}
=\frac{1000}{\pi^{\frac{1}{3}}\cdot 2^{-\frac{2}{3}}\cdot 1000^{\frac{2}{3}}}
=\frac{2\cdot 1000^{\frac{1}{3}}}{(2\pi)^{\frac{1}{3}}}
=\frac{20}{\sqrt[3] {2\pi} }=\frac{10\sqrt[3] {4\pi^2} }{\pi}=10\sqrt[3]{\frac{4}{\pi}}$$
