% Vendredi 28 Mai 2004 à 04:36:53

\opset{decimalsepsymbol={,}}
\newcommand\dfrac[2]{{\displaystyle\frac{#1}{#2}}}
\begin{center}
  \fbox{Règles sur les puissances}
\end{center}
\begin{enumerate}
\item $a^{-m}=\left(\dfrac{1}{a}\right)^m$.
  
  Exemple :
  \opcopy{4}{a}%
  \opcopy{2}{m}%
  \opneg{m}{mm}%
  \oppower{a}{m}{p1}%
  \oppower{a}{mm}{p2}%
  \opunzero{p2}%
  $\opprint{a}^{\opprint{mm}} =
  \dfrac{1}{\opprint{a}^{\opprint{m}}} =
  \dfrac{1}{\opprint{p1}} = \opprint{p2}$
  
\item $a^m \times a^n = a^{m+n}$.
  
  Exemple :
  \opcopy{3}{a}%
  \opcopy{4}{m}%
  \opcopy{5}{n}%
  \oppower{a}{m}{am}%
  \oppower{a}{n}{an}%
  \opadd*{m}{n}{m+n}%
  \oppower{a}{m+n}{a(m+n)}%
  $\opprint{a}^{\opprint{m}} \times
  \opprint{a}^{\opprint{n}} =
  \opprint{am} \times \opprint{an} = \opprint{a(m+n)}$
  et $\opprint{a}^{\opprint{m+n}} = \opprint{a(m+n)}$
  
\item $\dfrac{a^m}{a^n} = a^{m-n}$.
  
  Exemple :
  \opcopy{2}{a}%
  \opcopy{5}{m}%
  \opcopy{2}{n}%
  \oppower{a}{m}{am}%
  \oppower{a}{n}{an}%
  \opsub*{m}{n}{m-n}
  \oppower{a}{m-n}{a(m-n)}%
  $\dfrac{\opprint{a}^{\opprint{m}}}
         {\opprint{a}^{\opprint{n}}} =
  \dfrac{\opprint{am}}{\opprint{an}} = \opprint{a(m-n)}$
  et $\opprint{a}^{\opprint{m}-\opprint{n}} =
  \opprint{a}^{\opprint{m-n}} = \opprint{a(m-n)}$

\item $a^m \times b^m = (ab)^m$.

  Exemple :
  \opcopy{3}{a}%
  \opcopy{4}{b}%
  \opcopy{3}{m}%
  \oppower{a}{m}{am}%
  \oppower{b}{m}{bm}%
  \opmul*{a}{b}{ab}%
  \oppower{ab}{m}{abm}%
  $\opprint{a}^{\opprint{m}} \times
  \opprint{b}^{\opprint{m}} =
  \opprint{am} \times \opprint{bm} =
  \opprint{abm}$
  et $(\opprint{a} \times \opprint{b})^{\opprint{m}} =
  \opprint{ab}^{\opprint{m}} = \opprint{abm}$
\end{enumerate}