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.

More generally, if $a + b \sqrt{2}$ (with $a$, $b$ integers) has a cube root in $\mathbb Z[\sqrt{2}]$, the norm $N(a + b \sqrt{2}) = a^2 - 2 b^2 \sqrt{2}$ must be the cube of an integer, say $m^3$, and then that cube root would be
$x + y \sqrt{2}$ with $x^2 - 2 y^2 = m$.  From $(x+y \sqrt{2})^3 = a + b \sqrt{2}$ we get $x^3 + 6 x y^2 = 4 x^3 - 3 m x = a$.  In particular, $x$ must be a divisor of $a$, which brings you down to a finite (and hopefully not too big) list of possiblities that are easily checked.
