Let \(K\) be a field and \(A, B\) and \(C=A\otimes_KB\) be unital commutative Noetherian \(K\)-algebras, with proper ideals \(I<A, J<B\). The authors look into the problem of relationship between powers of \(I+J\) and powers of \(I\) and \(J\). They construct primary decompositions for powers of \(I+J\) from filtered primary components for powers of \(I\) and \(J\) in the case where \(K\) is algebraically closed. The author’s are also interested in ideals \(L\) in a commutative ring \(R\) that satisfy the persistence property on associated primes, namely such that \(\mathrm{Ass}(R/L^)\leq\mathrm{Ass}(R/L^n)\), for every \(n>0\). If \(J\) is a non-zero normal ideal (i.e. all powers of \(J\) are integrally closed), then \(\mathrm{Ass}(C/(I+J)^{n-1})\leq\mathrm{Ass}(C/(I+J)^n)\). Associated primes of each power of \(I+J\) are determined, in terms of the associated powers of \(I\) and \(J\). The paper is peppered with a number of technical lemmas and the authors acknowledge extensive use of results by H. T. Ha, D. H. Nguyen, N. V. Trung, T. N. Trung [“Symbolic powers of sums of ideals”, Preprint, arXiv:1702.01766].