Simplifying expression, is it possible? To show that, 
$$
(7 + 50^{1/2})^{1/3} + (7 - 50^{1/2})^{1/3} = 2
$$
I am aware of the way where we can "guess" and come up with the following:
7 + 50^(1/2) = (1 + 2^(1/2))^3
7 - 50^(1/2) = (1 - 2^(1/2))^3

Hence simplifying the expression. But can we do it without the guesswork?

We can use also the following identity.
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc).$$
Let $a=\sqrt[3]{7-5\sqrt2},$ $b=\sqrt[3]{7+5\sqrt2}$ and $c=\sqrt[3]{7-5\sqrt2}+\sqrt[3]{7+5\sqrt2}$.
Thus, since $a+b-c=0,$ we obtain:
$$7-5\sqrt2+7+5\sqrt2-c^3+3\sqrt[3]{(7-5\sqrt2)(7+5\sqrt2)}c=0$$ or
$$c^3+3c-14=0$$ or
$$c^3-2c^2+2c^2-4c+7c-14=0$$ or
$$(c-2)(c^2+2c+7)=0$$ or
$$c=2$$ and we are done!
