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Question Number 138366 by KwesiDerek last updated on 12/Apr/21

If x^2 +x^(−2) =(√(2+(√(2+(√2)))))  x^(16) +x^(−16) =?  Any help

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

Answered by TheSupreme last updated on 14/Apr/21

x^4 +1−(√(2+(√(2+(√2)))))x^2 =0  x^2 =t and solve for t  than evaluate  t^4 +(1/t^4 )

$${x}^{\mathrm{4}} +\mathrm{1}−\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{x}^{\mathrm{2}} =\mathrm{0} \\ $$$${x}^{\mathrm{2}} ={t}\:{and}\:{solve}\:{for}\:{t} \\ $$$${than}\:{evaluate} \\ $$$${t}^{\mathrm{4}} +\frac{\mathrm{1}}{{t}^{\mathrm{4}} } \\ $$

Answered by Ñï= last updated on 14/Apr/21

x^2 +x^(−2) =(√((x^2 +x^(−2) )^2 ))=(√(2+x^4 +x^(−4) ))                  =(√(2+(√((x^4 +x^(−4) )^2 ))))=(√(2+(√(2+x^8 +x^(−8) ))))                  =(√(2+(√(2+(√((x^8 +x^(−8) )^2 ))))))=(√(2+(√(2+(√(2+x^(16) +x^(−16) ))))))  ⇒x^(16) +x^(−16) =0

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

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