Quantitative Reasoning

So we're given that x^(a-b) = 1. Well, if x to the "something" = 1, there are only three ways that that can be true (I'm not even looking at the statements yet). Either x must itself equal 1 -- in which case the "something" can equal whatever in the world it wants to, since 1^whatever will still always equal 1 -- or x must equal -1, since -1 raised to any even "something" will equal 1 -- or "something" must equal 0, in which case x can equal whatever it wants to (other than 0), since anything^0 = 1.

The first statement -- a does not equal b -- tells us that "something" does not equal 0. That means that in order to make the given true, x itself must equal 1 or -1 (since those were the two other possibilities).

The second statement -- x > 0 -- doesn't independently guarantee anything about x, since if a-b = 0, then x can take any value it wants (in this case any value greater than 0).

When we combine them, though, we know that x is -1 or +1, AND that x>0, so x MUST BE 1.