Hello,

I'm working on a proof about Cantor Normal Form. In our notes we have that $\displaystyle \alpha \omega = \omega ^{\gamma_1} \omega$ and as part of the proof I think I need to find a similar equation of the form $\displaystyle \alpha^{n-1} \omega = .. $

Can anyone suggest how I would find finish this equation?

Just to put it in perspective, this is the question I'm doing:

Let $\displaystyle \alpha$ be a limit ordinal with Cantor Normal Form given by $\displaystyle \alpha = \omega^{\gamma_1} n_1 + \omega^{\gamma_2} n_2 + ... + \omega^{\gamma_k} n_k$ and $\displaystyle n < \omega$. Prove that $\displaystyle \alpha^n = \omega^{\gamma_1(n-1)} \alpha $.

EDIT: sorry I should specify that $\displaystyle \alpha$, $\displaystyle \omega$ and $\displaystyle \gamma$ are ordinals and $\displaystyle n$ is a positive integer.