Let $(R,{\frak m})$ be a local ring with prime ideals ${\frak p}$ and ${\frak q}$ such that $\sqrt{{\frak p}+{\frak q}}={\frak m}$ . If $R$ is regular and contains a field, and $\dim(R/{\frak p})+\dim(R/{\frak q})=\dim(R)$ , then it is proved that ${\frak p}^{(m)}\cap {\frak q}^{(n)}\subseteq {\frak m}^{m+n}$ for all positive integers $m$ and $n$ . This is proved using a generalization of Serre's Intersection Theorem which is applied to a hypersurface $R/fR$ . The generalization gives conditions that guarantee that Serre's bound on the intersection dimension $\dim(R/{\frak p})+\dim(R/{\frak q})\le \dim(R)$ holds when $R$ is nonregular.