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Prove or disprove that there is a rational number x and an irrational number y such that x^{y}is irrational.

The book works out the case with x and y irrational and x

^{y}rational. They used the nonconstructive existence proof method with x = sqrt(2) and y = sqrt(2). If that's rational, then you're finished. If it's irrational, then you can simply raise it to the power of sqrt(2) to get 2. I'm not sure how to adapt this approach to this problem. If it is irrational, then you're finished. If it's rational, then I'm not sure how to show you manipulate it to be irrational.

Suppose that a and b are odd integers with [itex]a\neq{b}.[/itex] Show there is a unique integer c such that |a - c| = |b - c|.

I know this is an easy problem, but I'm stumped for some reason. I've tried to find a general expression for c. The uniqueness method is appropriate here. I first need to show that an integer c with that property does exist. I then need to show that it's unique.