Bunuel wrote:
If u > 0 and v > 0, which is greater, \(u^v\) or \(v^u\)?
(1) u = 1
(2) v > 2
Solution:
Question Stem Analysis:
We need to determine which of u^v and v^u is greater, given that both u and v are positive.
Statement One Alone:
Since u = 1, u^v = 1^v = 1 (1 raised to any power is 1) and v^u = v^1 = v (any number raised to the power of 1 is itself). However, we can’t determine which one is greater. For example, if v = 0.5, then u^v > v^u since 1 > 0.5. However, if v = 2, then v^u > u^v since 2 > 1. Statement one alone is not sufficient.
Statement Two Alone:
Knowing only that v > 2 is not sufficient to determine whether u^v > v^u or v^u > u^v. For example, if u = 1, and v = 3, then u^v = 1^3 = 1 and v^u = 3^1 = 3. However, if u = 3 and v = 3, then both u^v and v^u are equal to 27. Statement two alone is not sufficient.
Statements One and Two Together:From statement one, we see that u^v = 1 and v^u = v. From statement two, we see that v > 2, so v^u > 2, and hence it’s greater than u^v. Both statements together are sufficient.
Answer: C _________________