L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(-\dfrac{53}{5}\right)\sqrt{9}+\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{9}\right)+\left(\dfrac{4}{5}\right)\sqrt{12}-3+\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(\left(\dfrac{13}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{12}\right)\) et \( Y=\left(-4\right)\sqrt{75}\) . 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{79}{9}\right)\sqrt{12}+\left(-\dfrac{53}{5}\right)\sqrt{9}+\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{9}\right)+\left(\dfrac{4}{5}\right)\sqrt{12}-3+\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(\left(\dfrac{13}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{12}\right)\right)+\left(\left(-4\right)\sqrt{75}\right)\\
&=&\left(\left(-\dfrac{158}{9}\right)\sqrt{3}-\dfrac{159}{5}+\left(\left(16\right)\sqrt{3}\right)-\dfrac{51}{2}-\left(\left(-10\right)\sqrt{3}\right)+1+\left(\dfrac{8}{5}\right)\sqrt{3}-3+\left(-\dfrac{158}{9}\right)\sqrt{3}+\left(\left(\dfrac{13}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{3}\right)\right)+\left(\left(-20\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{158}{9}\right)\sqrt{3}-\dfrac{159}{5}+\left(\left(16\right)\sqrt{3}\right)-\dfrac{51}{2}-\left(\left(-10\right)\sqrt{3}\right)+1+\left(\dfrac{8}{5}\right)\sqrt{3}-3+\left(-\dfrac{158}{9}\right)\sqrt{3}+\left(\left(\dfrac{13}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{3}\right)+\left(-20\right)\sqrt{3}\\
&=&\left(\dfrac{1237}{45}\right)\sqrt{3}-\dfrac{593}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(-\dfrac{53}{5}\right)\sqrt{9}+\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{9}\right)+\left(\dfrac{4}{5}\right)\sqrt{12}-3+\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(\left(\dfrac{13}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{12}\right)\right)-\left(\left(-4\right)\sqrt{75}\right)\\
&=&\left(\left(-\dfrac{158}{9}\right)\sqrt{3}-\dfrac{159}{5}+\left(\left(16\right)\sqrt{3}\right)-\dfrac{51}{2}-\left(\left(-10\right)\sqrt{3}\right)+1+\left(\dfrac{8}{5}\right)\sqrt{3}-3+\left(-\dfrac{158}{9}\right)\sqrt{3}+\left(\left(\dfrac{13}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{3}\right)\right)-\left(\left(-20\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{2137}{45}\right)\sqrt{3}-\dfrac{593}{10}\right)-\left(\left(-20\right)\sqrt{3}\right)\\
&=&\left(\dfrac{2137}{45}\right)\sqrt{3}-\dfrac{593}{10}+\left(20\right)\sqrt{3}\\
&=&\left(\dfrac{3037}{45}\right)\sqrt{3}-\dfrac{593}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(-\dfrac{53}{5}\right)\sqrt{9}+\left(\left(8\right)\sqrt{12}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{9}\right)-\left(\left(-5\right)\sqrt{12}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{9}\right)+\left(\dfrac{4}{5}\right)\sqrt{12}-3+\left(-\dfrac{79}{9}\right)\sqrt{12}+\left(\left(\dfrac{13}{6}\right)\sqrt{12}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{12}\right)\right)\times\left(\left(-4\right)\sqrt{75}\right)\\
&=&\left(\left(-\dfrac{158}{9}\right)\sqrt{3}-\dfrac{159}{5}+\left(\left(16\right)\sqrt{3}\right)-\dfrac{51}{2}-\left(\left(-10\right)\sqrt{3}\right)+1+\left(\dfrac{8}{5}\right)\sqrt{3}-3+\left(-\dfrac{158}{9}\right)\sqrt{3}+\left(\left(\dfrac{13}{3}\right)\sqrt{3}\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{3}\right)\right)\times\left(\left(-20\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{2137}{45}\right)\sqrt{3}-\dfrac{593}{10}\right)\left(\left(-20\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{8548}{9}\right)\sqrt{9}+\left(1186\right)\sqrt{3}\\
&=&-\dfrac{8548}{3}+\left(1186\right)\sqrt{3}\\
\end{eqnarray*}