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Question Number 162003 by mathlove last updated on 25/Dec/21

((27((27((27.... ))^(1/4) ))^(1/4) ))^(1/4) =x  (√(5(√(5(√(5(√(5.....))))))))=y  y^2 −x^2 =?

$$\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}....\:}}}={x} \\ $$$$\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}.....}}}}={y} \\ $$$${y}^{\mathrm{2}} −{x}^{\mathrm{2}} =? \\ $$

Answered by Rasheed.Sindhi last updated on 25/Dec/21

((27((27((27.... ))^(1/4) ))^(1/4) ))^(1/4) =x  (√(5(√(5(√(5(√(5.....))))))))=y  y^2 −x^2 =?  •x^4 =27((27((27.... ))^(1/4) ))^(1/4)  =27x ⇒x^3 =27  ⇒x=3⇒x^2 =9  •y^2 =5(√(5(√(5(√(5.....)))))) =5y⇒y=5    ⇒y^2 =25  y^2 −x^2 =25−9=16

$$\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}....\:}}}={x} \\ $$$$\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}.....}}}}={y} \\ $$$${y}^{\mathrm{2}} −{x}^{\mathrm{2}} =? \\ $$$$\bullet{x}^{\mathrm{4}} =\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}....\:}}\:=\mathrm{27}{x}\:\Rightarrow{x}^{\mathrm{3}} =\mathrm{27} \\ $$$$\Rightarrow{x}=\mathrm{3}\Rightarrow{x}^{\mathrm{2}} =\mathrm{9} \\ $$$$\bullet{y}^{\mathrm{2}} =\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}.....}}}\:=\mathrm{5}{y}\Rightarrow{y}=\mathrm{5} \\ $$$$\:\:\Rightarrow{y}^{\mathrm{2}} =\mathrm{25} \\ $$$${y}^{\mathrm{2}} −{x}^{\mathrm{2}} =\mathrm{25}−\mathrm{9}=\mathrm{16} \\ $$

Answered by aleks041103 last updated on 25/Dec/21

x=((27((27...))^(1/4) ))^(1/4) =((27x))^(1/4) ⇒x^4 =27x⇒x=3  y=(√(5(√(5(√(5...))))))=(√(5y))⇒y^2 =5y⇒y=5  y^2 −x^2 =5^2 −3^2 =16  Ans. 16

$${x}=\sqrt[{\mathrm{4}}]{\mathrm{27}\sqrt[{\mathrm{4}}]{\mathrm{27}...}}=\sqrt[{\mathrm{4}}]{\mathrm{27}{x}}\Rightarrow{x}^{\mathrm{4}} =\mathrm{27}{x}\Rightarrow{x}=\mathrm{3} \\ $$$${y}=\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}...}}}=\sqrt{\mathrm{5}{y}}\Rightarrow{y}^{\mathrm{2}} =\mathrm{5}{y}\Rightarrow{y}=\mathrm{5} \\ $$$${y}^{\mathrm{2}} −{x}^{\mathrm{2}} =\mathrm{5}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} =\mathrm{16} \\ $$$${Ans}.\:\mathrm{16} \\ $$

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