  Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real
field in which restricted power functions are definable. We show that if
$\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the
expansion $\widetilde{\mathbb R}^{\mathbb R}$ of $\widetilde{\mathbb R}$ by all
real power functions and an expansion (again in the sense of definability) of
$\widetilde{\mathbb R}$, then, provided that $\widetilde{\mathbb R}$ and
$\widehat{\mathbb R}$ have the same field of exponents, they define the same
sets. This can be viewed as a polynomially bounded version of an old conjecture
of van den Dries and Miller.
