How to show that $5/(1+2^{2/3}) = 1+2\sqrt[3]{2}-2^{2/3}$? I was trying to simplify the solution, $\frac{1}{5}(1+2\sqrt[3]{2}-2^{2/3})$, from a problem I done and saw that mathematica suggested $\frac{1}{1+2^{2/3}}$. I was wondering if anyone could show how to approach simplification of such expressions manually (it appears to amount to the step in the title of the post)?

This is the cubic formula from basic algebra:
$$a^3+b^3=(a+b)(a^2-ab+b^2)$$
$$\frac{a^3+b^3}{a+b}=a^2-ab+b^2$$
Observe,
$$\begin{align}(1+2^{2/3})×(1+2\sqrt[3]{2}-2^{2/3})&=1^3+2^2\\
&=5.\end{align}$$
