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Question Number 34408 by Rio Mike last updated on 05/May/18
what is the value of    (√(5(√(5(√(5(√(5(√(5(√5)))))))))))  ignoring 64.
$${what}\:{is}\:{the}\:{value}\:{of}\: \\ $$$$\:\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}}}}} \\ $$$${ignoring}\:\mathrm{64}. \\ $$
Commented by candre last updated on 06/May/18
x=(√(5(√(5(√(5(√(5(√(5(√5)))))))))))  x^2 =5(√(5(√(5(√(5(√(5(√5)))))))))  x^4 =5^3 (√(5(√(5(√(5(√5)))))))  x^8 =5^7 (√(5(√(5(√5)))))  x^(16) =5^(15) (√(5(√5)))  x^(32) =5^(31) (√5)  x^(64) =5^(63)   x=5^((63)/(64))
$${x}=\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}}}}} \\ $$$${x}^{\mathrm{2}} =\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}}}} \\ $$$${x}^{\mathrm{4}} =\mathrm{5}^{\mathrm{3}} \sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}}} \\ $$$${x}^{\mathrm{8}} =\mathrm{5}^{\mathrm{7}} \sqrt{\mathrm{5}\sqrt{\mathrm{5}\sqrt{\mathrm{5}}}} \\ $$$${x}^{\mathrm{16}} =\mathrm{5}^{\mathrm{15}} \sqrt{\mathrm{5}\sqrt{\mathrm{5}}} \\ $$$${x}^{\mathrm{32}} =\mathrm{5}^{\mathrm{31}} \sqrt{\mathrm{5}} \\ $$$${x}^{\mathrm{64}} =\mathrm{5}^{\mathrm{63}} \\ $$$${x}=\mathrm{5}^{\frac{\mathrm{63}}{\mathrm{64}}} \\ $$

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