x * exp(A*x) = B if A,B nonzero algebraic, has only transcendental roots x. Proof (which must have been known to RC Schroeppel): if x algebraic, then A*x is, hence exp(A*x) isn't, hence B isn't, except B is, hence x transcendental. Extension: And now if we take the ln() of both sides ln(x) = ln(B) - A*x is transcendental if B=1. And... exp(A*x) = B*x^P similarly has transcendental roots x, and ln(x) is transcendental, for any nonzero rational power P. In particular exp(x) = -x has a transcendent real root x = -0.567143 and then x, exp(x)=-x, exp(exp(x))=1/x, all transcendent.