if-x-2-x-2-2-2-2-x-16-x-16-

Question Number 138367 by KwesiDerek last updated on 12/Apr/21

if x^{2} + x^{- 2} = \sqrt{2 + \sqrt{2 + \sqrt{2}}}

x^{16} + x^{- 16} = ?

Commented by Kamel last updated on 12/Apr/21

x i s a c o m p l e x n u m b e r (4 s o l u t i o n s)

Answered by Kamel last updated on 12/Apr/21

Answered by cherokeesay last updated on 12/Apr/21

{(x^{2} + \frac{1}{x^{2}})}^{2} = 2 + \sqrt{2 + \sqrt{2}} \Leftrightarrow

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

{(x^{4} + \frac{1}{x^{4}})}^{2} = 2 + \sqrt{2} \Leftrightarrow x^{8} + \frac{1}{x^{8}} + 2 = 2 + \sqrt{2}

{(x^{8} + \frac{1}{x^{8}})}^{2} = 2 \Leftrightarrow x^{16} + \frac{1}{x^{16}} + 2 = 2 \Rightarrow

x^{16} + \frac{1}{x^{16}} = 0