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Question Number 9141 by tawakalitu last updated on 20/Nov/16

Find the value of x if  ((√(2 + (√3))))^x  + ((√(2 − (√3))))^x  = 4

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{if} \\ $$$$\left(\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{3}}}\right)^{\mathrm{x}} \:+\:\left(\sqrt{\mathrm{2}\:−\:\sqrt{\mathrm{3}}}\right)^{\mathrm{x}} \:=\:\mathrm{4} \\ $$

Commented by tawakalitu last updated on 20/Nov/16

please help.

$$\mathrm{please}\:\mathrm{help}. \\ $$

Commented by RasheedSoomro last updated on 21/Nov/16

x=2

$$\mathrm{x}=\mathrm{2} \\ $$

Commented by tawakalitu last updated on 21/Nov/16

please working

$$\mathrm{please}\:\mathrm{working} \\ $$

Commented by sou1618 last updated on 21/Nov/16

(√(2+(√3)))=(1/(√(2−(√3))))  so  ⇔((√(2+(√3))))^x +((√(2−(√3))))^(−x) =4  t=((√(2+(√3))))^x   ⇒t+(1/t)=4  t^2 −4t+1=0  t=2±(√3)=2+(√3) , (1/(2+(√3)))  x=2,−2

$$\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}=\frac{\mathrm{1}}{\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}} \\ $$$${so} \\ $$$$\Leftrightarrow\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\right)^{{x}} +\left(\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\right)^{−{x}} =\mathrm{4} \\ $$$${t}=\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\right)^{{x}} \\ $$$$\Rightarrow{t}+\frac{\mathrm{1}}{{t}}=\mathrm{4} \\ $$$${t}^{\mathrm{2}} −\mathrm{4}{t}+\mathrm{1}=\mathrm{0} \\ $$$${t}=\mathrm{2}\pm\sqrt{\mathrm{3}}=\mathrm{2}+\sqrt{\mathrm{3}}\:,\:\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$$${x}=\mathrm{2},−\mathrm{2} \\ $$$$ \\ $$

Commented by tawakalitu last updated on 21/Nov/16

Thanks so much sir. God bless you.

$$\mathrm{Thanks}\:\mathrm{so}\:\mathrm{much}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

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