The value of sqrt{2 sqrt[3]{4 sqrt{2 sqrt[3]{ | vedclass

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(A) Let $x = \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}}$.Squaring both sides,we get $x^{2} = 2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}$.Now,cubi

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$2$

Vedclass · Answer

(A) Let $x = \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}}$.Squaring both sides,we get $x^{2} = 2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}$.Now,cubing both sides,we get $(x^{2})^{3} = 2^{3} 	imes 4 	imes \sqrt{2 \sqrt[3]{4 \ldots}}$.This simplifies to $x^{6} = 8 	imes 4 	imes x$.$x^{6} = 32x$.Since $x 
eq 0$,we divide by $x$ to get $x^{5} = 32$.$x^{5} = 2^{5}$,which implies $x = 2$.

The value of $\sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}}}}$ is

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