Are you messing with basics?

Number Theory Level 2

⋯ 256 ⏟ 60 × = 2 8 x \large \underbrace{\sqrt{\sqrt{\sqrt{\cdots\sqrt{256}}}}}_{60 \times \sqrt{\ }} = 2^{8^x}

What is x x ?

Note: Please don't put in guesses and not to use a calculator. This question is meant to challenge your basics.

-21 16 -19 -23 -17

Jake Tricole by Jake Tricole , 36, Taiwan

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1 solution

Joshua Lowrance
Sep 6, 2019

⋯ 256 = 25 6 1 2 60 = ( 2 8 ) 1 2 60 = 2 8 ∗ 1 2 60 = 2 2 3 ∗ 1 2 60 = 2 1 2 57 = 2 ( 2 − 57 ) = 2 ( 2 ( 3 ∗ − 19 ) ) = 2 ( 8 − 19 ) \sqrt{\sqrt{\cdots \sqrt{\sqrt{256}}}} = 256^{\frac{1}{2^{60}}} = (2^8)^{\frac{1}{2^{60}}} = 2^{8*\frac{1}{2^{60}}} = 2^{2^3*\frac{1}{2^{60}}} = 2^{\frac{1}{2^{57}}} = 2^{(2^{-57})} = 2^{(2^{(3*-19)})} = 2^{(8^{-19})}

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