  We show that the limiting eigenvalue density of the product of n identically
distributed random matrices from an isotropic unitary ensemble (IUE) is equal
to the eigenvalue density of n-th power of a single matrix from this ensemble,
in the limit when the size of the matrix tends to infinity. Using this
observation one can derive the limiting density of the product of n independent
identically distributed non-hermitian matrices with unitary invariant measures.
In this paper we discuss two examples: the product of n Girko-Ginibre matrices
and the product of n truncated unitary matrices. We also provide an evidence
that the result holds also for isotropic orthogonal ensembles (IOE).
