  There are two different notions for symbolic powers of ideals existing in the
literature, one defined in terms of associated primes, the other in terms of
minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and
Trung, we interpret both notions of symbolic powers as suitable saturations of
the ordinary powers. We prove a binomial expansion formula for saturated powers
of sums of ideals. This gives a uniform treatment to an array of existing and
new results on both notions of symbolic powers of such sums: binomial expansion
formulas, computations of depth and regularity, and criteria for the equality
of ordinary and symbolic powers.
