L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-1\right)\sqrt{18}+\left(-\dfrac{79}{3}\right)\sqrt{50}+\left(\dfrac{71}{9}\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}-\left(\left(-\dfrac{19}{5}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{50}\right)\) et \( Y=\left(\left(\left(3\right)\sqrt{8}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}+\left(-\dfrac{49}{4}\right)\sqrt{8}+\left(\dfrac{49}{8}\right)\sqrt{4}+\dfrac{19}{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-1\right)\sqrt{18}+\left(-\dfrac{79}{3}\right)\sqrt{50}+\left(\dfrac{71}{9}\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}-\left(\left(-\dfrac{19}{5}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{50}\right)\right)+\left(\left(\left(\left(3\right)\sqrt{8}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}+\left(-\dfrac{49}{4}\right)\sqrt{8}+\left(\dfrac{49}{8}\right)\sqrt{4}+\dfrac{19}{4}\right)\right)\\
&=&\left(\left(-3\right)\sqrt{2}+\left(-\dfrac{395}{3}\right)\sqrt{2}+\left(\dfrac{142}{9}\right)\sqrt{2}+\left(-31\right)\sqrt{2}+\left(-20\right)\sqrt{2}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}+\dfrac{38}{5}-\left(\left(25\right)\sqrt{2}\right)\right)+\left(\left(\left(\left(6\right)\sqrt{2}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}+\left(-\dfrac{49}{2}\right)\sqrt{2}+\dfrac{49}{4}+\dfrac{19}{4}\right)\right)\\
&=&\left(-3\right)\sqrt{2}+\left(-\dfrac{395}{3}\right)\sqrt{2}+\left(\dfrac{142}{9}\right)\sqrt{2}+\left(-31\right)\sqrt{2}+\left(-20\right)\sqrt{2}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}+\dfrac{38}{5}-\left(\left(25\right)\sqrt{2}\right)+\left(\left(\left(6\right)\sqrt{2}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}+\left(-\dfrac{49}{2}\right)\sqrt{2}+\dfrac{49}{4}+\dfrac{19}{4}\right)\\
&=&\left(-\dfrac{16339}{126}\right)\sqrt{2}-\dfrac{14327}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-1\right)\sqrt{18}+\left(-\dfrac{79}{3}\right)\sqrt{50}+\left(\dfrac{71}{9}\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}-\left(\left(-\dfrac{19}{5}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{50}\right)\right)-\left(\left(\left(\left(3\right)\sqrt{8}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}+\left(-\dfrac{49}{4}\right)\sqrt{8}+\left(\dfrac{49}{8}\right)\sqrt{4}+\dfrac{19}{4}\right)\right)\\
&=&\left(\left(-3\right)\sqrt{2}+\left(-\dfrac{395}{3}\right)\sqrt{2}+\left(\dfrac{142}{9}\right)\sqrt{2}+\left(-31\right)\sqrt{2}+\left(-20\right)\sqrt{2}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}+\dfrac{38}{5}-\left(\left(25\right)\sqrt{2}\right)\right)-\left(\left(\left(\left(6\right)\sqrt{2}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}+\left(-\dfrac{49}{2}\right)\sqrt{2}+\dfrac{49}{4}+\dfrac{19}{4}\right)\right)\\
&=&\left(\left(-\dfrac{1754}{9}\right)\sqrt{2}-\dfrac{3679}{60}\right)-\left(\left(\dfrac{913}{14}\right)\sqrt{2}-\dfrac{329}{18}\right)\\
&=&\left(-\dfrac{1754}{9}\right)\sqrt{2}-\dfrac{3679}{60}+\left(-\dfrac{913}{14}\right)\sqrt{2}+\dfrac{329}{18}\\
&=&\left(-\dfrac{32773}{126}\right)\sqrt{2}-\dfrac{7747}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-1\right)\sqrt{18}+\left(-\dfrac{79}{3}\right)\sqrt{50}+\left(\dfrac{71}{9}\right)\sqrt{8}+\left(-\dfrac{31}{5}\right)\sqrt{50}+\left(-4\right)\sqrt{50}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}-\left(\left(-\dfrac{19}{5}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{50}\right)\right)\times\left(\left(\left(\left(3\right)\sqrt{8}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}+\left(-\dfrac{49}{4}\right)\sqrt{8}+\left(\dfrac{49}{8}\right)\sqrt{4}+\dfrac{19}{4}\right)\right)\\
&=&\left(\left(-3\right)\sqrt{2}+\left(-\dfrac{395}{3}\right)\sqrt{2}+\left(\dfrac{142}{9}\right)\sqrt{2}+\left(-31\right)\sqrt{2}+\left(-20\right)\sqrt{2}+7-\dfrac{67}{6}-\dfrac{67}{2}-\dfrac{79}{4}-\dfrac{23}{2}+\dfrac{38}{5}-\left(\left(25\right)\sqrt{2}\right)\right)\times\left(\left(\left(\left(6\right)\sqrt{2}\right)-\dfrac{52}{9}+\dfrac{9}{2}\right)-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}+\left(-\dfrac{49}{2}\right)\sqrt{2}+\dfrac{49}{4}+\dfrac{19}{4}\right)\right)\\
&=&\left(\left(-\dfrac{1754}{9}\right)\sqrt{2}-\dfrac{3679}{60}\right)\left(\left(\dfrac{913}{14}\right)\sqrt{2}-\dfrac{329}{18}\right)\\
&=&\left(-\dfrac{800701}{63}\right)\sqrt{4}+\left(-\dfrac{9901789}{22680}\right)\sqrt{2}+\dfrac{1210391}{1080}\\
&=&-\dfrac{183695503}{7560}+\left(-\dfrac{9901789}{22680}\right)\sqrt{2}\\
\end{eqnarray*}