I'm struggling to understand this isomorphism: $\mathbb{Z}_p^{\times}\simeq(\mathbb{Z}/p)^{\times}\times\mathbb{Z}_p$

Question

I'm starting to learn about $\mathbb{Z}_p$-extensions and looking at extensions $\mathbb{Q}(\zeta_{p^\infty})$ which have galois group $\mathbb{Z}_p^{\times}$ over the rationals. To get a $\mathbb{Z}_p$-extension, there's an isomorphism $\mathbb{Z}_p^{\times}\simeq(\mathbb{Z}/p)^{\times}\times\mathbb{Z}_p$ that allows us to drop to the galois group of the field fixed by $(\mathbb{Z}/p)^{\times}$ over the rationals. I'm struggling to understand this isomorphism. Could I get some help understanding this?