# Math Help - Cantor Normal Form

1. ## Cantor Normal Form

Hello,

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

Can anyone suggest how I would find finish this equation?

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

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

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

2. Hi

Use the associativity of ordinal multiplication!
If $n>1,\ \alpha^{n-1}\omega=\alpha^{n-2}(\alpha\omega)=...$