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 $x^2 + \frac{ 1}{x^2} ?$

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