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

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

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

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

Answered by aleks041103 last updated on 25/Dec/21

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

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