L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{40}{7}\right)\sqrt{12}\right)-\dfrac{67}{4}-\left(\left(-2\right)\sqrt{27}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{12}\right)+\left(-\dfrac{49}{6}\right)\sqrt{27}\) et \( Y=\left(\left(\left(-\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{71}{8}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{64}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)\right)\) . 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{40}{7}\right)\sqrt{12}\right)-\dfrac{67}{4}-\left(\left(-2\right)\sqrt{27}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{12}\right)+\left(-\dfrac{49}{6}\right)\sqrt{27}\right)+\left(\left(\left(\left(-\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{71}{8}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{64}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{80}{7}\right)\sqrt{3}\right)-\dfrac{67}{4}-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{3}\right)+\left(-\dfrac{49}{2}\right)\sqrt{3}\right)+\left(\left(\left(\left(-\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{355}{8}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{128}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{80}{7}\right)\sqrt{3}\right)-\dfrac{67}{4}-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{3}\right)+\left(-\dfrac{49}{2}\right)\sqrt{3}+\left(\left(\left(-\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{355}{8}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{128}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{11089}{840}\right)\sqrt{3}-\dfrac{67}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{40}{7}\right)\sqrt{12}\right)-\dfrac{67}{4}-\left(\left(-2\right)\sqrt{27}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{12}\right)+\left(-\dfrac{49}{6}\right)\sqrt{27}\right)-\left(\left(\left(\left(-\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{71}{8}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{64}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{80}{7}\right)\sqrt{3}\right)-\dfrac{67}{4}-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{3}\right)+\left(-\dfrac{49}{2}\right)\sqrt{3}\right)-\left(\left(\left(\left(-\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{355}{8}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{128}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{1565}{42}\right)\sqrt{3}-\dfrac{67}{4}\right)-\left(\left(\dfrac{6737}{280}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{1565}{42}\right)\sqrt{3}-\dfrac{67}{4}+\left(-\dfrac{6737}{280}\right)\sqrt{3}\\
&=&\left(-\dfrac{51511}{840}\right)\sqrt{3}-\dfrac{67}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{40}{7}\right)\sqrt{12}\right)-\dfrac{67}{4}-\left(\left(-2\right)\sqrt{27}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{12}\right)+\left(-\dfrac{49}{6}\right)\sqrt{27}\right)\times\left(\left(\left(\left(-\dfrac{9}{4}\right)\sqrt{75}\right)-\left(\left(\dfrac{71}{8}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{75}\right)-\left(\left(\left(\dfrac{64}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{27}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)-\left(\left(\dfrac{32}{7}\right)\sqrt{12}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{80}{7}\right)\sqrt{3}\right)-\dfrac{67}{4}-\left(\left(-6\right)\sqrt{3}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{3}\right)+\left(-\dfrac{49}{2}\right)\sqrt{3}\right)\times\left(\left(\left(\left(-\dfrac{45}{4}\right)\sqrt{3}\right)-\left(\left(\dfrac{355}{8}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{225}{2}\right)\sqrt{3}\right)-\left(\left(\left(\dfrac{128}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{3}\right)\right)\right)\\
&=&\left(\left(-\dfrac{1565}{42}\right)\sqrt{3}-\dfrac{67}{4}\right)\left(\left(\dfrac{6737}{280}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2108681}{2352}\right)\sqrt{9}+\left(-\dfrac{451379}{1120}\right)\sqrt{3}\\
&=&-\dfrac{2108681}{784}+\left(-\dfrac{451379}{1120}\right)\sqrt{3}\\
\end{eqnarray*}