Closure of exponentiation of real algebraic numbers.

Given two real, nonzero algebraic numbers a and b, with a > 0 (so that it excludes complex numbers), is there any named subset of the reals S such that (a^{b) belongs to S forall a,b? I know it's not all the reals since there should be countably many a}b's, since a,b are also countable.