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Applied Combinatorics

Section B.9 Exponentiation

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*}