If-x-1-2-find-the-value-of-x-4-1-x-4-

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

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

Answered by liberty last updated on 06/Nov/20

x^{2} = 1 + \sqrt{2} \Rightarrow x^{4} = 3 + 2 \sqrt{2}

\Leftrightarrow x^{4} + \frac{1}{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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