  We continue a very fruitful line of inquiry into the multiplicative ideal
theory of an arbitrary Leavitt path algebra L. Specifically, we show that
factorizations of an ideal in L into irredundant products or intersections of
finitely many prime-power ideals are unique, provided that the ideals involved
are powers of distinct prime ideals. We also characterize the completely
irreducible ideals in L, which turn out to be prime-power ideals of a special
type, as well as ideals that can be factored into products or intersections of
finitely many completely irreducible ideals.
