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Which is greater, $3^{5^3}$ or $6^{2^6}$?

 Jan 31, 2021
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3^5^3 is equal to 3^125

6^2^6 is equal to 6^64

6^64 is equal to 3^64 times 2^64.

3^125 is equal to 3^64 times 3^61.

So we find whether 2^64 or 3^61 is greater.

3^61 is greater than 2^61 by a factor of (3/2)^61.

2 has 3 powers remaining.

So the question is whether 8(2^3) or (3/2)^61 is greater.

1.5^7 is 11.

(3/2)^61 is greater so the first one is greater.

 Jan 31, 2021

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