<s>[INST] Prove that $(2+ \sqrt5)^{\frac13} + (2- \sqrt5)^{\frac13}$ is an integer When checked in calculator it is 1. But how to prove it?
Also it is not a normal addition like $x+ \frac1x$ which needs direct rationalization. So I just need something to proceed.
 [/INST] We have $x- \left(2+\sqrt 5 \right)^{\frac{1}{3}} - \left(2-\sqrt 5 \right)^{\frac{1}{3}} = 0$
So $x^3 -(2-\sqrt 5)-(2+\sqrt 5) = 3x \left(2+\sqrt 5 \right)^{\frac{1}{3}}\left(2-\sqrt 5 \right)^{\frac{1}{3}}= -3x$
(invoking that $a^3+b^3+c^3 = 3abc$ when $a+b+c = 0$)
from which we see that $x$ is a root of $x^3+3x-4 =0$ 
Since the derivative is positive, this means it has only one real root, which by inspection is $x=1$
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