  We investigate the rational powers of ideals. We find that in the case of
monomial ideals, the canonical indexing leads to a characterization of the
rational powers yielding that symbolic powers of squarefree monomial ideals are
indeed rational powers themselves. Using the connection with symbolic powers
techniques, we use splittings to show the convergence of depths and normalized
Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for
rational powers, and as a consequence of this we show the before-now unknown
convergence of Stanley depths of integral closure powers. In addition, we show
the finiteness of asymptotic associated primes, and we find that the normalized
lengths of local cohomology modules converge for rational powers, and hence for
symbolic powers of squarefree monomial ideals.
