L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{67}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{12}+\left(\dfrac{1}{6}\right)\sqrt{12}+\left(-\dfrac{37}{6}\right)\sqrt{12}-\dfrac{64}{3}\right)\) et \( Y=-\dfrac{1}{3}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{12}+\left(\dfrac{1}{6}\right)\sqrt{12}+\left(-\dfrac{37}{6}\right)\sqrt{12}-\dfrac{64}{3}\right)\right)+\left(-\dfrac{1}{3}\right)\\
&=&\left(\left(\left(-\dfrac{134}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{3}+\left(\dfrac{1}{3}\right)\sqrt{3}+\left(-\dfrac{37}{3}\right)\sqrt{3}-\dfrac{64}{3}\right)\right)+\left(-\dfrac{1}{3}\right)\\
&=&\left(\left(-\dfrac{134}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{3}+\left(\dfrac{1}{3}\right)\sqrt{3}+\left(-\dfrac{37}{3}\right)\sqrt{3}-\dfrac{64}{3}\right)-\dfrac{1}{3}\\
&=&\left(-\dfrac{14}{3}\right)\sqrt{3}+21\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{12}+\left(\dfrac{1}{6}\right)\sqrt{12}+\left(-\dfrac{37}{6}\right)\sqrt{12}-\dfrac{64}{3}\right)\right)-\left(-\dfrac{1}{3}\right)\\
&=&\left(\left(\left(-\dfrac{134}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{3}+\left(\dfrac{1}{3}\right)\sqrt{3}+\left(-\dfrac{37}{3}\right)\sqrt{3}-\dfrac{64}{3}\right)\right)-\left(-\dfrac{1}{3}\right)\\
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{64}{3}\right)-\left(-\dfrac{1}{3}\right)\\
&=&\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{64}{3}+\dfrac{1}{3}\\
&=&\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{65}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{12}\right)-\left(\left(-\dfrac{34}{3}\right)\sqrt{12}+\left(\dfrac{1}{6}\right)\sqrt{12}+\left(-\dfrac{37}{6}\right)\sqrt{12}-\dfrac{64}{3}\right)\right)\times\left(-\dfrac{1}{3}\right)\\
&=&\left(\left(\left(-\dfrac{134}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{16}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{68}{3}\right)\sqrt{3}+\left(\dfrac{1}{3}\right)\sqrt{3}+\left(-\dfrac{37}{3}\right)\sqrt{3}-\dfrac{64}{3}\right)\right)\times\left(-\dfrac{1}{3}\right)\\
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{3}+\dfrac{64}{3}\right)\left(-\dfrac{1}{3}\right)\\
&=&\left(\dfrac{14}{9}\right)\sqrt{3}-\dfrac{64}{9}\\
&=&\left(\dfrac{14}{9}\right)\sqrt{3}-\dfrac{64}{9}\\
\end{eqnarray*}