algebra precalculus - 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?

Answer

$$\frac{1000}{\pi \cdot (\frac{500}{\pi})^{\frac{2}{3}}}=$$

$$2 \frac {\frac {500}{\pi}}{(\frac {500}{\pi})^\frac {2}{3}}=$$

$$2(\frac {500}{\pi})^\frac {1}{3}=$$

$$10(\frac {4}{\pi})^\frac {1}{3}$$