  We produce an infinite family of transcendental numbers which, when raised to
their own power, become rational. We extend the method, to investigate positive
rational solutions to the equation $x^x = \alpha$, where $\alpha$ is a fixed
algebraic number. We then explore the consequences of $x^{P(x)}$ being
rational, if $x$ is rational and $P(x)$ is a fixed integer polynomial.
