  We study the images of the complex Ginibre eigenvalues under the power maps
$\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following
equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M
{\rm{Gin}} (N,M,k), $$ where the so-called Power-Ginibre distributions
${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. The
decomposition can be extended to any radially symmetric normal matrix ensemble,
and generalizes Rains' superposition theorem for the CUE and Kostlan's
independence of radii to a wider class of point processes. Our proof technique
also allows us to recover a result by Edelman and La Croix for the GUE.
  Concerning the Power-Ginibre blocks, we prove convergence of fluctuations of
their smooth linear statistics to independent gaussian variables, coherent with
the link between the complex Ginibre Ensemble and the Gaussian Free Field.
  Finally, some partial results about two-dimensional beta ensembles with
radial symmetry and even parameter $\beta$ are discussed, replacing
independence by conditional independence.
