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Algebra Difficulty 4.9 AIME Find the answer

Compute 22222⋯5432 \sqrt{2 \sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}}

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Taking the base 2 logarithm of the expression gives 1+12(1+13(1+14(1+⋯ )))=1+12!+13!+14!+⋯=e−11+\frac{1}{2}\left(1+\frac{1}{3}\left(1+\frac{1}{4}(1+\cdots)\right)\right)=1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots=e-1 Therefore the expression is just 2e−12^{e-1}.

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