L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(-\dfrac{43}{4}\right)\sqrt{75}+\left(-\dfrac{49}{3}\right)\sqrt{12}\) et \( Y=\left(\dfrac{9}{8}\right)\sqrt{9}+\left(8\right)\sqrt{27}-\dfrac{51}{2}+\left(2\right)\sqrt{12}+\dfrac{61}{3}-\dfrac{51}{2}+7\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(-\dfrac{43}{4}\right)\sqrt{75}+\left(-\dfrac{49}{3}\right)\sqrt{12}\right)+\left(\left(\dfrac{9}{8}\right)\sqrt{9}+\left(8\right)\sqrt{27}-\dfrac{51}{2}+\left(2\right)\sqrt{12}+\dfrac{61}{3}-\dfrac{51}{2}+7\right)\\
&=&\left(\left(-23\right)\sqrt{3}+\left(-\dfrac{215}{4}\right)\sqrt{3}+\left(-\dfrac{98}{3}\right)\sqrt{3}\right)+\left(\dfrac{27}{8}+\left(24\right)\sqrt{3}-\dfrac{51}{2}+\left(4\right)\sqrt{3}+\dfrac{61}{3}-\dfrac{51}{2}+7\right)\\
&=&\left(-23\right)\sqrt{3}+\left(-\dfrac{215}{4}\right)\sqrt{3}+\left(-\dfrac{98}{3}\right)\sqrt{3}+\dfrac{27}{8}+\left(24\right)\sqrt{3}-\dfrac{51}{2}+\left(4\right)\sqrt{3}+\dfrac{61}{3}-\dfrac{51}{2}+7\\
&=&\left(-\dfrac{977}{12}\right)\sqrt{3}-\dfrac{487}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(-\dfrac{43}{4}\right)\sqrt{75}+\left(-\dfrac{49}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{9}{8}\right)\sqrt{9}+\left(8\right)\sqrt{27}-\dfrac{51}{2}+\left(2\right)\sqrt{12}+\dfrac{61}{3}-\dfrac{51}{2}+7\right)\\
&=&\left(\left(-23\right)\sqrt{3}+\left(-\dfrac{215}{4}\right)\sqrt{3}+\left(-\dfrac{98}{3}\right)\sqrt{3}\right)-\left(\dfrac{27}{8}+\left(24\right)\sqrt{3}-\dfrac{51}{2}+\left(4\right)\sqrt{3}+\dfrac{61}{3}-\dfrac{51}{2}+7\right)\\
&=&\left(\left(-\dfrac{1313}{12}\right)\sqrt{3}\right)-\left(-\dfrac{487}{24}+\left(28\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{1313}{12}\right)\sqrt{3}+\dfrac{487}{24}+\left(-28\right)\sqrt{3}\\
&=&\left(-\dfrac{1649}{12}\right)\sqrt{3}+\dfrac{487}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(-\dfrac{43}{4}\right)\sqrt{75}+\left(-\dfrac{49}{3}\right)\sqrt{12}\right)\times\left(\left(\dfrac{9}{8}\right)\sqrt{9}+\left(8\right)\sqrt{27}-\dfrac{51}{2}+\left(2\right)\sqrt{12}+\dfrac{61}{3}-\dfrac{51}{2}+7\right)\\
&=&\left(\left(-23\right)\sqrt{3}+\left(-\dfrac{215}{4}\right)\sqrt{3}+\left(-\dfrac{98}{3}\right)\sqrt{3}\right)\times\left(\dfrac{27}{8}+\left(24\right)\sqrt{3}-\dfrac{51}{2}+\left(4\right)\sqrt{3}+\dfrac{61}{3}-\dfrac{51}{2}+7\right)\\
&=&\left(\left(-\dfrac{1313}{12}\right)\sqrt{3}\right)\left(-\dfrac{487}{24}+\left(28\right)\sqrt{3}\right)\\
&=&\left(\dfrac{639431}{288}\right)\sqrt{3}+\left(-\dfrac{9191}{3}\right)\sqrt{9}\\
&=&\left(\dfrac{639431}{288}\right)\sqrt{3}-9191\\
\end{eqnarray*}