L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-4\right)\sqrt{18}\right)+\left(\left(\dfrac{74}{7}\right)\sqrt{8}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-8\right)\sqrt{50}\right)+\left(\left(5\right)\sqrt{8}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{50}\right)-\dfrac{32}{7}\) et \( Y=\left(\dfrac{11}{2}\right)\sqrt{50}-\dfrac{33}{2}+\left(-\dfrac{51}{2}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-4\right)\sqrt{18}\right)+\left(\left(\dfrac{74}{7}\right)\sqrt{8}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-8\right)\sqrt{50}\right)+\left(\left(5\right)\sqrt{8}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{50}\right)-\dfrac{32}{7}\right)+\left(\left(\dfrac{11}{2}\right)\sqrt{50}-\dfrac{33}{2}+\left(-\dfrac{51}{2}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-12\right)\sqrt{2}\right)+\left(\left(\dfrac{148}{7}\right)\sqrt{2}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-40\right)\sqrt{2}\right)+\left(\left(10\right)\sqrt{2}\right)-\left(\left(\dfrac{85}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)+\left(\left(\dfrac{55}{2}\right)\sqrt{2}-\dfrac{33}{2}+\left(-\dfrac{153}{2}\right)\sqrt{2}\right)\\
&=&\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-12\right)\sqrt{2}\right)+\left(\left(\dfrac{148}{7}\right)\sqrt{2}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-40\right)\sqrt{2}\right)+\left(\left(10\right)\sqrt{2}\right)-\left(\left(\dfrac{85}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}+\left(\dfrac{55}{2}\right)\sqrt{2}-\dfrac{33}{2}+\left(-\dfrac{153}{2}\right)\sqrt{2}\\
&=&-\dfrac{8301}{280}+\left(-\dfrac{117}{14}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-4\right)\sqrt{18}\right)+\left(\left(\dfrac{74}{7}\right)\sqrt{8}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-8\right)\sqrt{50}\right)+\left(\left(5\right)\sqrt{8}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{50}\right)-\dfrac{32}{7}\right)-\left(\left(\dfrac{11}{2}\right)\sqrt{50}-\dfrac{33}{2}+\left(-\dfrac{51}{2}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-12\right)\sqrt{2}\right)+\left(\left(\dfrac{148}{7}\right)\sqrt{2}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-40\right)\sqrt{2}\right)+\left(\left(10\right)\sqrt{2}\right)-\left(\left(\dfrac{85}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)-\left(\left(\dfrac{55}{2}\right)\sqrt{2}-\dfrac{33}{2}+\left(-\dfrac{153}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{3681}{280}+\left(\dfrac{569}{14}\right)\sqrt{2}\right)-\left(\left(-49\right)\sqrt{2}-\dfrac{33}{2}\right)\\
&=&-\dfrac{3681}{280}+\left(\dfrac{569}{14}\right)\sqrt{2}+\left(49\right)\sqrt{2}+\dfrac{33}{2}\\
&=&\dfrac{939}{280}+\left(\dfrac{1255}{14}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-4\right)\sqrt{18}\right)+\left(\left(\dfrac{74}{7}\right)\sqrt{8}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-8\right)\sqrt{50}\right)+\left(\left(5\right)\sqrt{8}\right)-\left(\left(\dfrac{17}{2}\right)\sqrt{50}\right)-\dfrac{32}{7}\right)\times\left(\left(\dfrac{11}{2}\right)\sqrt{50}-\dfrac{33}{2}+\left(-\dfrac{51}{2}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{66}{5}-\dfrac{66}{5}-\left(\left(-12\right)\sqrt{2}\right)+\left(\left(\dfrac{148}{7}\right)\sqrt{2}\right)-\dfrac{71}{5}+\dfrac{45}{8}-\left(\left(-40\right)\sqrt{2}\right)+\left(\left(10\right)\sqrt{2}\right)-\left(\left(\dfrac{85}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)\times\left(\left(\dfrac{55}{2}\right)\sqrt{2}-\dfrac{33}{2}+\left(-\dfrac{153}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{3681}{280}+\left(\dfrac{569}{14}\right)\sqrt{2}\right)\left(\left(-49\right)\sqrt{2}-\dfrac{33}{2}\right)\\
&=&\left(-\dfrac{7401}{280}\right)\sqrt{2}+\dfrac{121473}{560}+\left(-\dfrac{3983}{2}\right)\sqrt{4}\\
&=&\left(-\dfrac{7401}{280}\right)\sqrt{2}-\dfrac{2109007}{560}\\
\end{eqnarray*}