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\lhead[]{\small 1^{ère} STI~GC2}
\chead[]{\textsc{Outils de calcul}}
\rhead[]{\small {Fiche n°3}}
\lfoot{\small \url{http://mathematiques.daval.free.fr}}
\rfoot{\small -\thepage-}
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\begin{document}
   \begin{center}
   \shadowbox{\begin{LARGE}
      FRACTIONS, PUISSANCES, RACINES CARRÉES
   \end{LARGE}}
\end{center}


\setlength{\parskip}{3ex plus 1ex minus 1ex}
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\section{Les fractions}

\Ovalbox{
\begin{minipage}{16.5cm}
\textbf{Rappels :} chaque dénominateur étant non nuls, on peut écrire : \\
\textcolor{blue}{
\begin{tabular}{C{1.4}lC{2.5}l}
& & & \\
& $\dfrac{a}{c}\pm\dfrac{b}{c} =\dfrac{a\pm b}{c}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\dfrac{2}{3}-\dfrac{7}{3} =\dfrac{2-7}{3} =\dfrac{-5}{3}$ \\
& & & \\
& $\dfrac{a}{b}\pm\dfrac{c}{d} =\dfrac{ad\pm bc}{bd}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\dfrac{3}{8}+\dfrac{2}{3} =\dfrac{3\times3}{8\times3}+\dfrac{2\times8}{3\times8} =\dfrac{9+16}{24} =\dfrac{25}{24}$ \\
& & & \\
& $\dfrac{a}{b}\times\dfrac{c}{d} =\dfrac{ac}{bd}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\dfrac{-3}{5}\times\dfrac{1}{7} = \dfrac{-3\times1}{5\times7} =\dfrac{-3}{35}$ \\
& & & \\
& $\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}} =\dfrac{a}{b}\times\dfrac{d}{c}=\dfrac{ad}{bc}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\dfrac{\dfrac{1}{4}}{\dfrac{7}{9}} =\dfrac{1}{4}\times\dfrac{9}{7}=\dfrac{9}{28}$ \\ [25pt]
\end{tabular}
}
\end{minipage}
}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Exo}
   \everymath={\displaystyle}
   Donner l'écriture des nombres suivants sous la forme d'un entier ou d'une fraction irréductible.
   \begin{multicols}{2}
      $A ={{1 \over 2} - {1 \over 3} + {1 \over 4}}$
      
      $B ={2 - {13 \over 7} + \left( 1 + {5 \over 2} \right)}$
      
      $C ={\left( {2\over3} - {3\over4} \right) + 
3\left( {4\over5} - {5\over6} \right)}$

      $D ={\left( {1\over2} + {5\over3} \right) \times\left( 3 + {7\over4} \right) : \left( {1\over2} - {5\over6} \right)}$

      $E =\dfrac{\dfrac23 + \dfrac34 - \dfrac45 \times \dfrac34}{\dfrac25 \times \dfrac43 + \dfrac13}$
   \end{multicols}
\end{Exo}


\section{Les puissances}

\Ovalbox{
\begin{minipage}{16.5cm}
\textbf{Rappels :} Pour tous réels $a$ et $b$ non nuls, $m$ et $n$ entiers, on peut écrire : \\
\textcolor{blue}{
\begin{tabular}{C{1.4}lC{2.5}l}
& & & \\
& $a^n\times a^m =a^{n+m}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $2^3\times 2^6 =2^{3+6} =2^9$ \\
& & & \\
&  $\dfrac{a^n}{a^m} =a^{n-m}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\dfrac{5^8}{5^2} =5^{8-2} =5^6$ \\
& & & \\
& $(a\times b)^n =a^n\times b^n$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $(3\times 7)^2 =3^2\times 7^2$ \\
& & & \\
& $(a^n)^m =a^{n\times n}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $(10^3)^5 =10^{3\times 5} =10^{15}$ \\ [10pt]
\end{tabular}
}
\end{minipage}
}

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Exo}
   \everymath={\displaystyle}
   Simplifier les nombres  suivants :
   \begin{multicols}{2}
      $A =3^2\times3^{-4}\times3^7\times3^$
      
      $B =\dfrac{2\times2^2\times2^3}{2^4\times2^5}$
      
      $C =(2\times3^2\times3^3)^4$
      
      $D ={2^3 \times 5^4\times 7^3 \over 5^3 \times 7^2 \times2}$
          
      $E =81^5 \times (3^{-2})^{-5} \times {1\over 9}$
      
      $F ={4^{-2}\times 8^3\over 16^3}$
      
      $G ={9^3\times 27^2\times 75\over 5^2\times 3^4}.
$
      
      $H =\left( {2\over 3}\right) ^{11}\times \left( {3\over2}\right) ^{10}$
      
      $I =(a^3)^2 \times a^{-4}$
      
      $J =a^2b^{-3} (ab)^4$
   \end{multicols}
\end{Exo}

\smallskip

\section{Les racines carrées}

\Ovalbox{
\begin{minipage}{16.5cm}
\textbf{Rappels :} pour tous réels $a$ et $b$ positifs, on peut écrire : \\
\textcolor{blue}{
\begin{tabular}{C{1.4}lC{2.5}l}
& & & \\
& $(\sqrt{a})^2 =\sqrt{a^2} =a$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $(\sqrt{3})^2 =\sqrt{9} =3$ \\
& & & \\
& $\sqrt{a\times b} =\sqrt{a}\times\sqrt{b}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\sqrt{4\times 3} =\sqrt{4}\times\sqrt{3} =2\sqrt{3}$ \\
& & & \\
& $\sqrt{\dfrac{a}{b}} =\dfrac{\sqrt{a}}{\sqrt{b}}$ & $\overset{\text{\tiny{exemple}}}{\Longrightarrow}}$ & $\sqrt{\dfrac{5}{9}} =\dfrac{\sqrt{5}}{\sqrt{9}} =\dfrac{\sqrt{5}}{3}$ \\
& & & \\
\begin{LARGE}\textcolor{red}{\danger}\end{LARGE} & \textcolor{red}{$\sqrt{a+b} \neq\sqrt{a}+\sqrt{b}$} & \textcolor{red}{$\overset{\text{\tiny{contrexemple}}}{\Longrightarrow}}$} & \textcolor{red}{$\sqrt{9+16} =\sqrt{25} =5$} \\
& & & \textcolor{red}{$\sqrt{9}+\sqrt{16} =3+4 =7$} \\ [10pt]
\end{tabular}
}
\end{minipage}
}

\bigskip

\begin{Exo}
   \everymath={\displaystyle}
   Simplifier l'écriture des nombres suivants :
   \begin{multicols}{2}
      $A =\sqrt{12}$
      
      $B =\sqrt{48}$
      
      $C =\sqrt{36+64}$
      
      $D =3\sqrt2+8\sqrt2-5\sqrt2$$
      
      $E =5\sqrt{27} - 2 \sqrt{48}$
      
      $F ={{\sqrt{81} \over \sqrt{242}} \times \sqrt{98 \over 25}}$   
      
      $G ={3 \over \sqrt7}$
      
      $H =(2-\sqrt{3})^2$
      
      $I =3(1+\sqrt{2})(1-\sqrt{2})$    
   \end{multicols}
\end{Exo}


  
\end{document}
