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

Algebra Level 3

If we are given that x+1x=a x + \frac{ 1}{x} = a , then we know that x2+1x2=a22 x^2 + \frac{ 1}{x^2} = a^2 - 2 and x3+1x3=a33a 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 x be a real number such that x3+1x3=102 x^3 + \frac{1}{x^3} = 10 \sqrt{2} , what is the value of x2+1x2? x^2 + \frac{ 1}{x^2} ?

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