Is $(ω^ω)^ω$ equal to $ω^{(ω^2)}$?

Question

I was just wondering if exponentiation rules such as in the title apply to transfinite ordinals... So things like:

$\omega^\omega \cdot \omega^\omega = \omega^{\omega2}$

Answer 1

Yes, $\alpha^{\beta}\cdot\alpha^\gamma = \alpha^{\beta+\gamma}$ and $(\alpha^\beta)^\gamma = \alpha^{\beta\cdot \gamma}$ hold generally for ordinals.