L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{47}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{4}+\left(-5\right)\sqrt{4}+\dfrac{19}{8}+\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{55}{6}\right)\sqrt{18}+\left(\dfrac{49}{4}\right)\sqrt{8}-9+\left(-1\right)\sqrt{4}\right)\) et \( Y=\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{50}\right)\) . 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{47}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{4}+\left(-5\right)\sqrt{4}+\dfrac{19}{8}+\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{55}{6}\right)\sqrt{18}+\left(\dfrac{49}{4}\right)\sqrt{8}-9+\left(-1\right)\sqrt{4}\right)\right)+\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(-\dfrac{47}{2}\right)\sqrt{2}\right)-\left(\dfrac{130}{7}-10+\dfrac{19}{8}+\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{2}+\left(\dfrac{49}{2}\right)\sqrt{2}-9-2\right)\right)+\left(-\dfrac{39}{2}-\left(\left(\dfrac{255}{2}\right)\sqrt{2}+\left(-5\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{47}{2}\right)\sqrt{2}\right)-\left(\dfrac{130}{7}-10+\dfrac{19}{8}+\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{2}+\left(\dfrac{49}{2}\right)\sqrt{2}-9-2\right)-\dfrac{39}{2}-\left(\left(\dfrac{255}{2}\right)\sqrt{2}+\left(-5\right)\sqrt{2}\right)\\
&=&\left(-131\right)\sqrt{2}-\dfrac{1089}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{47}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{4}+\left(-5\right)\sqrt{4}+\dfrac{19}{8}+\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{55}{6}\right)\sqrt{18}+\left(\dfrac{49}{4}\right)\sqrt{8}-9+\left(-1\right)\sqrt{4}\right)\right)-\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(-\dfrac{47}{2}\right)\sqrt{2}\right)-\left(\dfrac{130}{7}-10+\dfrac{19}{8}+\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{2}+\left(\dfrac{49}{2}\right)\sqrt{2}-9-2\right)\right)-\left(-\dfrac{39}{2}-\left(\left(\dfrac{255}{2}\right)\sqrt{2}+\left(-5\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{17}{2}\right)\sqrt{2}+\dfrac{3}{56}\right)-\left(-\dfrac{39}{2}+\left(-\dfrac{245}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{17}{2}\right)\sqrt{2}+\dfrac{3}{56}+\dfrac{39}{2}+\left(\dfrac{245}{2}\right)\sqrt{2}\\
&=&\left(114\right)\sqrt{2}+\dfrac{1095}{56}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{47}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{4}+\left(-5\right)\sqrt{4}+\dfrac{19}{8}+\left(-6\right)\sqrt{8}\right)-\left(\left(-\dfrac{55}{6}\right)\sqrt{18}+\left(\dfrac{49}{4}\right)\sqrt{8}-9+\left(-1\right)\sqrt{4}\right)\right)\times\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{2}\right)\sqrt{50}+\left(-1\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(-\dfrac{47}{2}\right)\sqrt{2}\right)-\left(\dfrac{130}{7}-10+\dfrac{19}{8}+\left(-12\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{2}+\left(\dfrac{49}{2}\right)\sqrt{2}-9-2\right)\right)\times\left(-\dfrac{39}{2}-\left(\left(\dfrac{255}{2}\right)\sqrt{2}+\left(-5\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{17}{2}\right)\sqrt{2}+\dfrac{3}{56}\right)\left(-\dfrac{39}{2}+\left(-\dfrac{245}{2}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2547}{16}\right)\sqrt{2}+\left(\dfrac{4165}{4}\right)\sqrt{4}-\dfrac{117}{112}\\
&=&\left(\dfrac{2547}{16}\right)\sqrt{2}+\dfrac{233123}{112}\\
\end{eqnarray*}