OFFICIAL SOLUTION FROM MANHATTANWhen comparing numbers raised to different powers, the first method is typically to try to break the bases down and find similarities. Here, though, the bases are all prime numbers already, and they’re all different! Now what?
As always, start with the most annoying feature: those decimals in the exponents. If you raise every number to the 10th power, you’ll still preserve the relative order of the 5 numbers. All you care about is finding the largest, and because you’re starting with 5 positive bases, the largest number will remain the largest even after raising each to the 10th power.
(A) 2^273
(B) 3^182
(C) 5^111
(D) 7^91
(E) 11^51
So, let’s see, what’s 2 raised to the 273 power? 2, 4, 8, 16, 32, 64, … just kidding.
Check out the exponents: are there any similarities? Any common factors?
Yes! 273 and 182 are both multiples of 91. Take the 91st root of (A), (B), and (D) so that you can compare just those three. Taking the 91st root is the same as dividing all three by 91:
(A) 2^273 = 23 = 8
(B) 3^182 = 32 = 9
(D) 7^91 = 71 = 7
Excellent! Among these three, then, answer (B) is the largest; you can eliminate (A) and (D).
As for the other two choices, compare them one at a time to one of the three you've already compared, looking for benchmarks. Take (C): 5^111. Are there any small powers of 5 close to a power of 2, 3, or 7? Yes: 52 = 25, while 33 = 27.
How do we use that? Well, 33 > 52. Scale up by taking both sides to the 60th power: 3^180 > 5^120. This inequality slips nicely between the relevant answer choices. If 3^180 > 5^120 then the larger 3^182 must be bigger than the smaller 5^111. Eliminate answer (C).
Try this tactic again with answer (E): 11^2 = 121 and 2^7 = 128. In this case, 2^7 > 112. The power of 2 in answer (A) is 273, which divided by 7 yields 39. So take both sides of the inequality to the 39th power:
2^273 > 1178, which is larger than 11^51. Answer (E) is not the largest either!
The correct answer is (B).
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