I have a question here on ordinal arithmetic: Suppose $\displaystyle \alpha + \beta = \omega$ ($\displaystyle \alpha, \beta$ not zero). What are $\displaystyle \alpha \beta , \alpha^\beta$?

Am I right in assuming that the sum of two finite ordinals cannot be an infinite ordinal? If so I figure $\displaystyle \beta = \omega $ and $\displaystyle \alpha$ can just be any finite ordinal. So $\displaystyle \alpha \beta = \omega$ and $\displaystyle \alpha^\beta = \omega$ (by two rules in my notes).

Alpha and beta cannot be the other way round as right cancellation does not work.

So I think this proof is sound if the first statement is true. If anyone can comment on what I've done that'd be neat.