How to show that $5/(1+2^{2/3}) = 1+2\sqrt[3]{2}-2^{2/3}$?

Question

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)?

Answer 1

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,

$$\begin{align}(1+2^{2/3})×(1+2\sqrt[3]{2}-2^{2/3})&=1^3+2^2\\ &=5.\end{align}$$