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

Algebra Level 2

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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