If x=2+2^{1/2}, then what is the value of x^{4}+16x^{-4}?

Explanation

Given x = 2+\sqrt{2} \implies x-2 = \sqrt{2}. Squaring both sides gives x^2 - 4x + 2 = 0 \implies x + \frac{2}{x} = 4. Squaring again gives x^2 + 4x^{-2} + 4 = 16 \implies x^2 + 4x^{-2} = 12. Squaring a third time yields x^4 + 16x^{-4} + 8 = 144 \implies x^4 + 16x^{-4} = 136.

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