The definition of ordinal exponentiation for successor ordinals is:
$\alpha^{\beta+1} = \alpha^\beta \cdot \alpha$.
In general (for all ordinals $\beta$, also limit ordinals), is it true that
$\alpha \cdot \alpha^\beta = \alpha^\beta \cdot \alpha$?
And if so, what is the best way to prove it?
On
No -- $\alpha\cdot \beta$ is the order type of $\beta$ many copies of $\alpha$ after each other.
This means that $2\cdot\omega$ is just $\omega$: an $\omega$-sequence of elements that come two at a time is the same as an $\omega$-sequence of single elements.
$$ 2\cdot \omega = 2+2+2+\cdots = (1+1)+(1+1)+\cdots = 1+1+1+1+\cdots = \omega$$
On the other hand $\omega\cdot 2$ is an entirely new ordinal, the order type of two copies of the natural numbers after each other.
$$ \omega\cdot 2 = (1+1+1+\cdots) + (1+1+1+\cdots) $$
No.
$$\omega\cdot\omega^\omega = \omega^{1+\omega} = \omega^\omega < \omega^{\omega + 1} = \omega^\omega\cdot\omega$$