Math Olympiad | Which is Larger? | Calculator NOT allowed

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[@vishalmishra3046]

*Numbers with larger exponent grow faster than numbers with larger base* . So most likely, 5^4441 < 4^5441 = 2^10882 but let's make sure that is indeed true. To compare powers of 5 with powers of 2 that are close to each other, note that 5^3 = 125 < 128 = 2^7. So, 5^4441 = 5^(3 x 4441/3) = 125^(1480+1/3) < 125^1500 AND 4^5441 = 2^10882 = 128^(10882/7 = 1554+4/7) > 128^1500 So, 5^5551 < 125^1500 < 128^1500 < 2^10882 = 4^5441, so 5^4441 < 4^5441 *More generally* A^ (mn) = (A^m)^n and B^(pq) = (B^p)^q, so if (A^m >= B^p) and (n >= q) then A^(mn) >= B^(pq). It's best to choose m and p, such that A^m is very close to B^p (as in this case 125 and 128)

[@Danieswors]

4^5441

[@michallesz2]

Without counting, you can see that the power shows us how many digits a given number can have. So the first number would have about 4,000 digits and the second about 5,000 digits, so the second number is bigger.