Several posters have correctly simplified the expression:
b(b^a - a^b)
But let's think further about what it would take for this expression to be ODD. For a product to be odd, both of the terms in that product need to be odd. So b has to be odd; if it's even the whole product would be even.
What would make b^a - a^b odd? Well given that b has to be odd we know that an odd number taken to any positive integer power will also be odd, so b^a is odd. That means that a would have to be even, because ODD - EVEN = ODD.
So the conditions that we need to get an odd product are:
b is odd
a is even
If either of these is violated, we'll get an even product.
1) This violates our second condition. We'd have either:
even(even - odd) = even
odd(odd - odd) = even
Sufficient
2) This violates our first condition. Since we're multiplying b by the terms in the parentheses, that product has to be even.
Sufficient
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
