 Solution

The recursive definition of ordinal addition is $\alpha +0=\alpha$, $\alpha +(\beta +1)=(\alpha +\beta )+1$, and $\alpha +\lambda ={\sup }_{\beta <\lambda }(\alpha +\beta )$ for a nonzero limit ordinal $\lambda$. The synthetic definition is the order type of the ordered disjoint union of a copy of $\alpha$ followed by a copy of $\beta$.

For the synthetic operation, adding the empty order does nothing, adding a successor appends one final point, and concatenation with a limit order is the increasing union of the initial concatenations with all its proper initial segments. Their order types have supremum ${\sup }_{\beta <\lambda }(\alpha +\beta )$. Thus the synthetic operation satisfies the recursive equations. Transfinite induction gives uniqueness of the recursively defined operation, proving equivalence.

The recursive definitions of ordinal multiplication and ordinal exponentiation areThe last equation in each line is for nonzero limit $\lambda$. In particular the convention $0^0=1$ belongs to this recursive definition.