## SectionB.9Exponentiation

We now define a binary operation called exponentiation which is defined only on those ordered pairs $$(m,n)$$ of natural numbers where not both are zero. The notation for exponentiation is non-standard. In books, it is written $$m^n$$ while the notations $$m**n\text{,}$$ $$m\wedge n$$ and $$\exp(m,n)$$ are used in-line. We will use the $$m^n$$ notation for the most part.

When $$m=0\text{,}$$ we set $$0^n=0$$ for all $$n\in\nonnegints$$ with $$n\neq0\text{.}$$ Now let $$m\neq0\text{.}$$ We define $$m^n$$ by (i) $$m^0=1$$ and (ii) $$m^{k+1}=mm^k\text{.}$$

Let $$m,n\in\nonnegints$$ with $$m\neq0\text{.}$$ Then $$m^{n+0}=m^n=m^n\,1=m^n\,m^0\text{.}$$ Now suppose that $$m^{n+k}=m^n\,m^k\text{.}$$ Then

\begin{equation*} m^{n+(k+1)}=m^{(n+k)+1}=m\,m^{n+k} = m(m^n\,m^k)=m^n(m\,m^k)=m^n\,m^{k+1}. \end{equation*}

Let $$m,n\in\nonnegints$$ with $$m\neq0\text{.}$$ Then $$(m^n)^0=1=m^0=m^{n0}\text{.}$$ Now suppose that $$(m^n)^k=m^{nk}\text{.}$$ Then

\begin{equation*} (m^n)^{k+1}=m^n(m^n)^k=m^n(m^{nk})=m^{n+nk}=m^{n(k+1)}. \end{equation*}