  We introduce and study coordinate-wise powers of subvarieties of
$\mathbb{P}^n$, i.e. varieties arising from raising all points in a given
subvariety of $\mathbb{P}^n$ to the $r$-th power, coordinate by coordinate.
This corresponds to studying the image of a subvariety of $\mathbb{P}^n$ under
the quotient of $\mathbb{P}^n$ by the action of the finite group
$\mathbb{Z}_r^{n+1}$. We determine the degree of coordinate-wise powers and
study their defining equations, particularly for hypersurfaces and linear
spaces. Applying these results, we compute the degree of the variety of
orthostochastic matrices and determine iterated dual and reciprocal varieties
of power sum hypersurfaces. We also establish a link between coordinate-wise
squares of linear spaces and the study of real symmetric matrices with a
degenerate eigenspectrum.
