Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

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

if x^2 +x^(−2) =(√(2+(√(2+(√2))))) x^(16) +x^(−16) =?

$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{−\mathrm{2}} =\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}} \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{16}} +\boldsymbol{\mathrm{x}}^{−\mathrm{16}} =? \\ $$

Commented by Kamel last updated on 12/Apr/21

x is a complex number (4 solutions)

$${x}\:{is}\:{a}\:{complex}\:{number}\:\left(\mathrm{4}\:{solutions}\right) \\ $$

Answered by Kamel last updated on 12/Apr/21

Answered by cherokeesay last updated on 12/Apr/21

(x^2 +(1/x^2 ))^2 = 2+(√(2+(√2) )) ⇔ x^4 +(1/x^4 ) + 2 = 2 + (√(2+(√2))) ⇒ (x^4 +(1/x^4 ))^2 = 2+ (√2) ⇔ x^8 +(1/x^8 ) + 2 = 2+(√2) (x^8 +(1/x^8 ))^2 = 2 ⇔ x^(16) + (1/x^(16) ) + 2 = 2 ⇒ x^(16) + (1/x^(16) ) = 0

$$\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}} \:=\:\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}\:}\:\Leftrightarrow \\ $$$${x}^{\mathrm{4}} +\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\:+\:\mathrm{2}\:=\:\mathrm{2}\:+\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}\:\Rightarrow\: \\ $$$$\left({x}^{\mathrm{4}} +\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\right)^{\mathrm{2}} \:=\:\mathrm{2}+\:\sqrt{\mathrm{2}}\:\Leftrightarrow\:{x}^{\mathrm{8}} +\frac{\mathrm{1}}{{x}^{\mathrm{8}} }\:+\:\mathrm{2}\:=\:\mathrm{2}+\sqrt{\mathrm{2}} \\ $$$$\left({x}^{\mathrm{8}} +\frac{\mathrm{1}}{{x}^{\mathrm{8}} }\right)^{\mathrm{2}} \:=\:\mathrm{2}\:\Leftrightarrow\:{x}^{\mathrm{16}} \:+\:\frac{\mathrm{1}}{{x}^{\mathrm{16}} }\:+\:\mathrm{2}\:=\:\mathrm{2}\:\Rightarrow \\ $$$${x}^{\mathrm{16}} \:+\:\frac{\mathrm{1}}{{x}^{\mathrm{16}} }\:=\:\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com