= 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> are
$$
\alpha\cdot0=0,\quad\alpha(\beta+1)=\alpha\beta+\alpha,\quad\alpha\lambda=\sup_{\beta<\lambda}\alpha\beta;
$$
$$
\alpha^0=1,\quad\alpha^{\beta+1}=\alpha^\beta\alpha,\quad\alpha^\lambda=\sup_{\beta<\lambda}\alpha^\beta.
$$
The last equation in each line is for nonzero limit $\lambda$. In particular the convention $0^0=1$ belongs to this recursive definition.
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