  We show how multiplier ideals can be used to obtain uniform multiplicative
bounds for certain families of ideals on a smooth complex algebraic variety. In
particular we prove a quick but rather surprising result about symbolic powers
of radical ideals on such a variety. Specifically, let I be a radical ideal
sheaf on a smooth variety X defining a reduced subscheme Z of X and suppose
that every irreducible component of Z has codimension at most e in X. Given an
integer m > 0 suppose that f is a function germ that vanishes to order at least
e.m at a general point of each irreducible component of Z . Then in fact f lies
in the m-th power I^m of I.
