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<h2 class="thm">Thorme</h2><div class="thm"> Si \( a \neq 0 \) et si \( m \) et \( n \) sont deux entiers relatifs, alors on a : 
     <div class="math">\(\begin{matrix} 
    a^m \times a^n & =  &a^{m+n} \\
     \frac{a^m}{a^n} & = & a^{m-n} \\
   \left( a^m \right)^n & = & a^{m \times n}
   \end{matrix})</div> </div>




	\def{integer a1=random(2..4)*randint(-1,1)}
	\def{integer m1=random(2..5)*randint(-1,1)}
	\def{integer n1=random(2..5)*randint(-1,1)}
	\def{integer mn1=pari((\m1)+(\n1))}
	\def{integer a2=random(2..4)*randint(-1,1)}
	\def{integer m2= random(2..5)*randint(-1,1)}
	\def{integer n2=random(2..5)*randint(-1,1)}
	\def{integer mn2=pari((\m2)-(\n2))}
	\def{integer a3=random(2..4)*randint(-1,1)}
	\def{integer m3= random(2..5)*randint(-1,1)}
	\def{integer n3=random(2..5)*randint(-1,1)}
    \def{integer mn3=pari((\m3)*(\n3))}
	\def{rational result1 = pari( (\a1)^(\m1+\n1))}
	\def{rational result2 = pari( (\a2)^(\m2-\n2))}
	\def{rational result3 = pari( (\a3)^(\m3*\n3))}

\def<table align=center>
<tr><td> A  </td>
  	<td> = </td>
  	<td>\if{\a1<0}{\( (\a1)^{\m1} \times (\a1)^{\n1} )}{\(\a1^{\m1} \times \a1^{\n1})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> B  </td>
  	<td> = </td>
  	<td>\if{\a2<0}{\(\frac{ (\a2)^{\m2}}{ (\a2)^{\n2} })}{\(\frac{\a2^{\m2}}{ \a2^{\n2}})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> C  </td>
  	<td> = </td>
  	<td>\if{\a3<0}{\( \left( (\a3)^{\m3}\right)^{\n3})}{\( \left(\a3^{\m3}\right)^{\n3})}</td>
</tr>

<tr><td> A   </td>
  	<td> = </td>
  	<td>\if{\a1<0}{\( (\a1)^{\m1+(\n1)} )}{\(\a1^{\m1+(\n1)})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> B   </td>
  	<td> = </td>
  	<td>\if{\a2<0}{\( (\a2)^{\m2-(\n2)} )}{\(\a2^{\m2-(\n2)})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> C   </td>
  	<td> = </td>
  	<td>\if{\a3<0}{\( (\a3)^{\m3*(\n3)}) }{\( \a3^{\m3*(\n3)} )}</td>
</tr>

<tr><td> A   </td>
  	<td> = </td>
  	<td>\if{\a1<0}{\( (\a1)^{\mn1} )}{\(\a1^{\mn1})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> B   </td>
  	<td> = </td>
  	<td>\if{\a2<0}{\( (\a2)^{\mn2} )}{\(\a2^{\mn2})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> C   </td>
  	<td> = </td>
  	<td>\if{\a3<0}{\( (\a3)^{\mn3}) }{\( \a3^{\mn3} )}</td>
</tr>

<tr><td> (A   </td>
  	<td> = </td>
  	<td>\( \result1 ))</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td>( B   </td>
  	<td> = </td>
  	<td>\( \result2 ))</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td>( C   </td>
  	<td> = </td>
  	<td>\( \result3 ))</td>
</tr>
</table>

<h2 class="thm">Exercice</h2><div class="thm">
<ul>
    <li>\exercise{module=H3/algebra/oefpuis.fr&cmd=new&exo=produitdiv&cmd=new}{Produit de puissance}  </li>
    <li>\exercise{module=H3/algebra/oefpuis.fr&cmd=new&exo=quotientdiv&cmd=new}{Quotient de puissance}  </li>
    <li>\exercise{module=H3/algebra/oefpuis.fr&cmd=new&exo=puispuissance&cmd=new}{Puissance de puissance}  </li>
    <li>\exercise{module=H3/algebra/oefpuis.fr&cmd=new&exo=quprindiv&cmd=new}{Une des rgles au hasard}  </li>
<ul/></div>












<h2 class="thm">Thorme</h2><div class="thm">Si \( a \neq 0 \) et \( b \neq 0 \), et si \( n \) est un entier relatif, alors : 
    <div class="math">\(\begin{matrix} 
    \left( a \times b \right)^{n} & = & a^n \times b^n \\
     \left( \frac{a}{b} \right)^{n} & = & \frac{a^n}{b^n}
        \end{matrix})</div> </div>




<h2 class="rem">Remarque</h2><div class="rem">
    Pourquoi doit-on avoir \( a \neq 0 \) ?
</div>





	\def{integer a1=random(2..6)*randint(-1,1)}
	\def{integer b1=random(2..6)*randint(-1,1)}
	\def{integer a2=random(2..6)*randint(-1,1)}
	\def{integer b2=random(2..6)*randint(-1,1)}
	\def{integer n1=random(2..5)*randint(-1,1)}
	\def{integer n2=random(2..5)*randint(-1,1)}
	\def{rational result1 = pari( (\a1*\b1)^(\n1))}
	\def{rational result2 = pari( (\a2/\b2)^(\n2))}
\def<table align=center>
<tr><td> A  </td>
  	<td> = </td>
  	<td>\if{\b1<0}{\( \left(\a1 \times (\b1) \right)^{\n1})}{\( \left(\a1 \times \b1 \right)^{\n1})}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> B  </td>
  	<td> = </td>
  	<td>\( \left( \frac{\a2}{\b2} \right)^{\n2} )</td>
</tr>

<tr><td> A   </td>
  	<td> = </td>
  	<td>\if{\a1<0}{\if{\b1<0}{\( (\a1)^{\n1} \times (\b1)^{\n1})}{\( (\a1)^{\n1} \times \b1^{\n1})}}{\if{\b1<0}{\( \a1^{\n1} \times (\b1)^{\n1})}{\( \a1^{\n1} \times \b1^{\n1})}}</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td> B   </td>
  	<td> = </td>
  	<td>\if{\a2<0}{\if{\b2<0}{\( \frac{(\a2)^{\n2}}{(\b2)^{\n2}})}{\( \frac{(\a2)^{\n2}}{ \b2^{\n2}})}}{\if{\b2<0}{\( \frac{\a2^{\n2}}{ (\b2)^{\n2}})}{\( \frac{\a2^{\n2}}{\b2^{\n2}})}}</td>
</tr>
<tr><td> (A   </td>
  	<td> = </td>
  	<td>\( \result1 ))</td>
	<td> &nbsp; &nbsp; &nbsp; </td>
	<td>( B   </td>
  	<td> = </td>
  	<td>\( \result2 ))</td>
</tr>
</table>








<h2 class="thm">Exercice</h2><div class="thm">
<ul>
    <li>\exercise{module=H2/algebra/oefpower.fr&cmd=new&exo=operation1&cmd=new}{Appliquer la premire formule}  </li>
    <li>\exercise{module=H2/algebra/oefpower.fr&cmd=new&exo=operation2&cmd=new}{Appliquer la deuxime formule}  </li>
<ul/></div>

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