## 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*}