1 to 2 and 1 to 3 is simple, but what about 3 to 2?

Algebra Level 3

If we are given that $x + \frac{ 1}{x} = a$ , then we know that $x^2 + \frac{ 1}{x^2} = a^2 - 2$ and $x^3 + \frac{ 1}{ x^3} = a^3 - 3a$ . However, it's not too clear how we can relate the latter 2 equations with each other.

Let $x$ be a real number such that $x^3 + \frac{1}{x^3} = 10 \sqrt{2}$ , what is the value of $x^2 + \frac{ 1}{x^2} ?$