L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)-\left(\left(9\right)\sqrt{4}\right)+\left(-8\right)\sqrt{8}+\left(-\dfrac{76}{9}\right)\sqrt{18}+\dfrac{71}{8}-\left(\left(-3\right)\sqrt{50}\right)\) et \( Y=\left(\left(-\dfrac{32}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)+0-\left(\left(-8\right)\sqrt{18}\right)+\left(9\right)\sqrt{18}+\left(9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\dfrac{51}{7}+\left(9\right)\sqrt{50}\) . 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{15}{2}\right)\sqrt{50}\right)-\left(\left(9\right)\sqrt{4}\right)+\left(-8\right)\sqrt{8}+\left(-\dfrac{76}{9}\right)\sqrt{18}+\dfrac{71}{8}-\left(\left(-3\right)\sqrt{50}\right)\right)+\left(\left(\left(-\dfrac{32}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)+0-\left(\left(-8\right)\sqrt{18}\right)+\left(9\right)\sqrt{18}+\left(9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\dfrac{51}{7}+\left(9\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)-18+\left(-16\right)\sqrt{2}+\left(-\dfrac{76}{3}\right)\sqrt{2}+\dfrac{71}{8}-\left(\left(-15\right)\sqrt{2}\right)\right)+\left(\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)+0-\left(\left(-24\right)\sqrt{2}\right)+\left(27\right)\sqrt{2}+\left(45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\dfrac{51}{7}+\left(45\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)-18+\left(-16\right)\sqrt{2}+\left(-\dfrac{76}{3}\right)\sqrt{2}+\dfrac{71}{8}-\left(\left(-15\right)\sqrt{2}\right)+\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)+0-\left(\left(-24\right)\sqrt{2}\right)+\left(27\right)\sqrt{2}+\left(45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\dfrac{51}{7}+\left(45\right)\sqrt{2}\\
&=&\left(\dfrac{2257}{18}\right)\sqrt{2}-\dfrac{103}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)-\left(\left(9\right)\sqrt{4}\right)+\left(-8\right)\sqrt{8}+\left(-\dfrac{76}{9}\right)\sqrt{18}+\dfrac{71}{8}-\left(\left(-3\right)\sqrt{50}\right)\right)-\left(\left(\left(-\dfrac{32}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)+0-\left(\left(-8\right)\sqrt{18}\right)+\left(9\right)\sqrt{18}+\left(9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\dfrac{51}{7}+\left(9\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)-18+\left(-16\right)\sqrt{2}+\left(-\dfrac{76}{3}\right)\sqrt{2}+\dfrac{71}{8}-\left(\left(-15\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)+0-\left(\left(-24\right)\sqrt{2}\right)+\left(27\right)\sqrt{2}+\left(45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\dfrac{51}{7}+\left(45\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{67}{6}\right)\sqrt{2}-\dfrac{73}{8}\right)-\left(\left(\dfrac{1028}{9}\right)\sqrt{2}+\dfrac{51}{7}\right)\\
&=&\left(\dfrac{67}{6}\right)\sqrt{2}-\dfrac{73}{8}+\left(-\dfrac{1028}{9}\right)\sqrt{2}-\dfrac{51}{7}\\
&=&\left(-\dfrac{1855}{18}\right)\sqrt{2}-\dfrac{919}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{15}{2}\right)\sqrt{50}\right)-\left(\left(9\right)\sqrt{4}\right)+\left(-8\right)\sqrt{8}+\left(-\dfrac{76}{9}\right)\sqrt{18}+\dfrac{71}{8}-\left(\left(-3\right)\sqrt{50}\right)\right)\times\left(\left(\left(-\dfrac{32}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)+0-\left(\left(-8\right)\sqrt{18}\right)+\left(9\right)\sqrt{18}+\left(9\right)\sqrt{50}+\left(9\right)\sqrt{18}+\dfrac{51}{7}+\left(9\right)\sqrt{50}\right)\\
&=&\left(\left(\left(\dfrac{75}{2}\right)\sqrt{2}\right)-18+\left(-16\right)\sqrt{2}+\left(-\dfrac{76}{3}\right)\sqrt{2}+\dfrac{71}{8}-\left(\left(-15\right)\sqrt{2}\right)\right)\times\left(\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)+0-\left(\left(-24\right)\sqrt{2}\right)+\left(27\right)\sqrt{2}+\left(45\right)\sqrt{2}+\left(27\right)\sqrt{2}+\dfrac{51}{7}+\left(45\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{67}{6}\right)\sqrt{2}-\dfrac{73}{8}\right)\left(\left(\dfrac{1028}{9}\right)\sqrt{2}+\dfrac{51}{7}\right)\\
&=&\left(\dfrac{34438}{27}\right)\sqrt{4}+\left(-\dfrac{60538}{63}\right)\sqrt{2}-\dfrac{3723}{56}\\
&=&\dfrac{3756535}{1512}+\left(-\dfrac{60538}{63}\right)\sqrt{2}\\
\end{eqnarray*}