ac.commutative algebra - Formal power series ring & completion

I encountered the following passage in Matsumura's Commutative Ring Theory :

A a Noetherian ring, $B=A[[x]]$ a formal power series ring. $Msubset B$ a maximal ideal, $mathfrak{m}=Mcap A$. Then $(B_{M})^{mbox{^}}=(A_{mathfrak{m}})mbox{^}[[x]]$, where ^ indicate $M$-adic and $mathfrak{m}$-adic completions, respectively.

It's not immediately clear to me why this is the case. How should I go about proving this?
Thanks!