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\begin{document}$${\mathbb {k}}$$\end{document} be a field, let A and B be polynomial rings over k\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {k}}$$\end{document}, and let S=A⊗kB\documentclass[12pt]{minimal}
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\begin{document}$$S= A \otimes _{\mathbb {k}}B$$\end{document}. Let I⊆A\documentclass[12pt]{minimal}
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\begin{document}$$I \subseteq A$$\end{document} and J⊆B\documentclass[12pt]{minimal}
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\begin{document}$$J \subseteq B$$\end{document} be monomial ideals. We establish a binomial expansion for rational powers of I+J⊆S\documentclass[12pt]{minimal}
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\begin{document}$$I+J \subseteq S$$\end{document} in terms of those of I and J. Particularly, for u∈Q+\documentclass[12pt]{minimal}
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\begin{document}$$u \in {\mathbb Q}_+$$\end{document}, we prove that (I+J)u=∑0≤ω≤u,ω∈QIωJu-ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (I+J)_u = \sum _{0 \le \omega \le u, \ \omega \in {\mathbb Q}} I_\omega J_{u-\omega }, \end{aligned}$$\end{document}and that the sum on the right-hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of I and J. We further give sufficient conditions for this formula to hold for the integral closures of powers of I+J\documentclass[12pt]{minimal}
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\begin{document}$$I+J$$\end{document} in terms of those of I and J. Under these conditions, we provide explicit formulas for the depth and regularity of (I+J)k¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{(I+J)^k}$$\end{document} in terms of those of powers of I and J.
