  We give a brief history of transcendental number theory, including Schanuel's
conjecture (S). Assuming (S), we prove that if z and w are complex numbers, not
0 or 1, with z^w and w^z algebraic, then z and w are either both rational or
both transcendental. A corollary is that if (S) is true, then we can find four
distinct transcendental positive real numbers x, y, s, t such that the three
numbers x^y=/=y^x and s^t=t^s are all integers. Another application (possibly
known) is that (S) implies the transcendence of the numbers
sqrt(2)^sqrt(2)^sqrt(2), i^i^i, and i^e^pi. We also prove that if (S) holds and
a^a^z=z, where a=/=0 is algebraic and z is irrational, then z is
transcendental.
