Exponentiation is Primitive Recursive

Theorem
The function $$\operatorname{exp}: \N^2 \to \N$$, defined as:
 * $$\operatorname{exp} \left({n, m}\right) = n^m$$

is primitive recursive‎.

Proof
We observe that:
 * $$\operatorname{exp} \left({n, 0}\right) = n^0 = 1$$

and that
 * $$\operatorname{exp} \left({n, m + 1}\right) = n^\left({m + 1}\right) = \left({n^m}\right) \times n = \operatorname{mult} \left({\operatorname{exp} \left({n, m}\right), n}\right)$$.

Thus $$\operatorname{exp}$$ is defined by primitive recursion from the primitive recursive function ‎$\operatorname{mult}$.

Hence the result.