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Question Number 121218 by ZiYangLee last updated on 06/Nov/20

$$\mathrm{If}x=\sqrt{1+\sqrt{2}},\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}{x}^4+\frac{1}{x^4}.$$

Answered by liberty last updated on 06/Nov/20

$${\mathrm{x}}^2=1+\sqrt{2}\Rightarrow {\mathrm{x}}^4=3+2\sqrt{2}$$$$\iff {\mathrm{x}}^4+\frac{1}{{\mathrm{x}}^4}=3+2\sqrt{2}+\frac{1}{3+2\sqrt{2}}$$$$=3+2\sqrt{2}+\frac{3-2\sqrt{2}}{9-8}=6$$

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