Calculate $\sqrt [3] {7+5\sqrt {2}}+\sqrt [3] {7-5\sqrt {2}}$

Question

i need help. I tried but I could not(real numbers).

Answer 1

Let $a$ and $b$ be the two terms of the sum. Note that they are both real numbers, and $a^3 + b^3 = 14$ while $ab = -1$.

$$a^3+b^3 = (a+b)(a^2-ab+b^2) = (a+b)((a+b)^2-3ab)$$

Write $x=a+b$ , so $14 = x(x^2+3)$ or $x^3 + 3x -14 =0$. This equation has only one real solution. Solve it and you are done.