Answer to POW #5:

Perhaps the simplest possible answer is a=-2 and b=-4. Another answer is a = (3/2)^2 = 9/4 and b = (3/2)^3 = 27/8.

In fact, the possible solutions include all pairs of numbers of the form a = (1+1/k)^k and b=(1+1/k)^(k+1) will work, where k is a positive integer. (The second solution listed above corresponds to k=2.) To determine this formula, begin by setting b = ra. Then consider a^(ra) = (ra)^a and solve for a. Then consider which values of r will make a a rational number.

Source: Angela Dunn, Mathematical Bafflers