How to simplify $\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}$ The answer is 2. But I want to learn how to simplify this expression without the use of calculator.

Frequently, numbers of this kind are obtained from Cardano's formula for the roots of a cubic equation $x^3+px+q=0$, that is,
$$
\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}
+
\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}
$$
so you have
$$
-\frac{q}{2}=7,
\qquad
\frac{q^2}{4}+\frac{p^3}{27}=50,
$$
that easily gives
$$
q=-14,\qquad p=3.
$$
Therefore the equation is
$$
x^3+3x-14=0
$$
which has a single real root, precisely $2$.
