Schanuel's conjecture and algebraic powers zʷ and wᶻ with z and w transcendental

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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 zʷ and wᶻ 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 xʸ=/=yˣ and sᵗ=tˢ are all integers. Another application (possibly known) is that (S) implies the transcendence of the numbers sqrt(2)ˢqrt(2)ˢqrt(2), iⁱⁱ, and iᵉᵖi. We also prove that if (S) holds and aᵃᶻ=z, where a=/=0 is algebraic and z is irrational, then z is transcendental.