Let $\alpha,\beta$ be ordinals. We define $\alpha^\beta$ as follows:

Some properties of exponentiation:

1. $0^\alpha=0$ if $\alpha>0$
2. $1^\alpha=1$
3. $\alpha^1=\alpha$
4. $\alpha^\beta\cdot \alpha^\gamma=\alpha^{\beta+\gamma}$
5. $(\alpha^\beta)^\gamma=\alpha^{\beta\cdot\gamma}$
6. For any ordinals $\alpha,\beta$ with $\alpha>0$ and $\beta>1$ , there exists a unique triple $(\gamma,\delta,\epsilon)$ of ordinals such that $$\alpha=\beta^\gamma\cdot \delta+\epsilon$$ where $0<\delta<\beta$ and $\epsilon<\beta^\delta$ .

All of these properties can be proved using transfinite induction.