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RE: Brainsteem Mathematics Challenges: Integer Powers
47 = 16384 is your answer. In my case it helps that I simply memorized powers up to a certain point, including the powers of 2 up through 1048576 (220).
For those who don't want to brute-force the answer either, what they can do is try to turn everything into prime factorizations. So for example, 47 = (22)^7 = 214 and 83 = (23)^3 = 29.
65 becomes 25 x 35, and from there it's a matter of comparing 25 (32) against 33 (27) to see that 65 > 38. However, 29 (512) > 35 (243), so 47 is still greater.
You're eventually left with comparing 47 against 56 and 74, and only needing to calculate those. That's 16384 vs. 15625 vs. 2401.
You memorised them! lol. Actually, doesn't surprise me - if you try the follow-up question, there is also a tricky step that those who work a lot with numbers will know what to do.
I dislike books that call them "tricks"; they are not tricks, just knowledge of how numbers work!