Give the inductive definition of ordinal exponentiation. Use it to show that $\alpha^{\beta} \leqslant \alpha^{\gamma}$ whenever $\beta \leqslant \gamma$ (for $\alpha \geqslant 1$ ), and also that $\alpha^{\beta}<\alpha^{\gamma}$ whenever $\beta<\gamma($ for $\alpha \geqslant 2)$.

Give an example of ordinals $\alpha$ and $\beta$ with $\omega<\alpha<\beta$ such that $\alpha^{\omega}=\beta^{\omega}$.

Show that $\alpha^{\beta+\gamma}=\alpha^{\beta} \alpha^{\gamma}$, for any ordinals $\alpha, \beta, \gamma$, and give an example to show that we need not have $(\alpha \beta)^{\gamma}=\alpha^{\gamma} \beta^{\gamma}$.

For which ordinals $\alpha$ do we have $\alpha^{\omega_{1}} \geqslant \omega_{1}$ ? And for which do we have $\alpha^{\omega_{1}} \geqslant \omega_{2}$ ? Justify your answers.

[You may assume any standard results not concerning ordinal exponentiation.]