I want to obtain the result of: $$\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}$$ Which turns out to be 1. Now, let's prettend we don't know what the result is. I solved it by stating $$\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}=z$$ Then by cubing the equation: $$4-3\biggr(\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2} \biggr)=z^3$$ $$ z^3+3z-4=0$$ Now, just by an inexplicable mysticism, the equation can be restated as: $$(z-1)(z^2+z+4)=0$$ Therefore, $z=1$, which is what I wanted to prove.
Are there another ways to solve this problem? I find this method quite impractical and not so elegant. I'm interested in ways to solve it that are MUCH simpler.
Thanks in advance!
Another way is to observe that $$\sqrt[3]{\sqrt{5}\pm 2}= \frac{1}{2}(\sqrt{5}\pm 1)$$