L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-9\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{50}+\left(\dfrac{71}{8}\right)\sqrt{8}+\left(-\dfrac{31}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{8}+\left(8\right)\sqrt{18}+\dfrac{19}{7}+\left(\dfrac{1}{4}\right)\sqrt{4}\right)-\dfrac{44}{7}\) et \( Y=\left(\left(\left(-\dfrac{53}{8}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)-\left(-\dfrac{32}{5}+\left(-\dfrac{17}{5}\right)\sqrt{50}\right)-\left(\left(8\right)\sqrt{8}+\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{18}\right)\right)\) . 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(-9\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{50}+\left(\dfrac{71}{8}\right)\sqrt{8}+\left(-\dfrac{31}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{8}+\left(8\right)\sqrt{18}+\dfrac{19}{7}+\left(\dfrac{1}{4}\right)\sqrt{4}\right)-\dfrac{44}{7}\right)+\left(\left(\left(\left(-\dfrac{53}{8}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)-\left(-\dfrac{32}{5}+\left(-\dfrac{17}{5}\right)\sqrt{50}\right)-\left(\left(8\right)\sqrt{8}+\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{18}\right)\right)\right)\\
&=&\left(-18-\left(\left(-\dfrac{175}{8}\right)\sqrt{2}+\left(\dfrac{71}{4}\right)\sqrt{2}-31\right)-\left(\left(35\right)\sqrt{2}+\left(24\right)\sqrt{2}+\dfrac{19}{7}+\dfrac{1}{2}\right)-\dfrac{44}{7}\right)+\left(\left(-\dfrac{53}{4}-\left(\left(0\right)\sqrt{2}\right)\right)-\left(-\dfrac{32}{5}+\left(-17\right)\sqrt{2}\right)-\left(\left(16\right)\sqrt{2}-\dfrac{1}{2}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(21\right)\sqrt{2}\right)\right)\right)\\
&=&-18-\left(\left(-\dfrac{175}{8}\right)\sqrt{2}+\left(\dfrac{71}{4}\right)\sqrt{2}-31\right)-\left(\left(35\right)\sqrt{2}+\left(24\right)\sqrt{2}+\dfrac{19}{7}+\dfrac{1}{2}\right)-\dfrac{44}{7}+\left(-\dfrac{53}{4}-\left(\left(0\right)\sqrt{2}\right)\right)-\left(-\dfrac{32}{5}+\left(-17\right)\sqrt{2}\right)-\left(\left(16\right)\sqrt{2}-\dfrac{1}{2}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(21\right)\sqrt{2}\right)\right)\\
&=&-\dfrac{57}{20}+\left(-\dfrac{53}{2}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-9\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{50}+\left(\dfrac{71}{8}\right)\sqrt{8}+\left(-\dfrac{31}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{8}+\left(8\right)\sqrt{18}+\dfrac{19}{7}+\left(\dfrac{1}{4}\right)\sqrt{4}\right)-\dfrac{44}{7}\right)-\left(\left(\left(\left(-\dfrac{53}{8}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)-\left(-\dfrac{32}{5}+\left(-\dfrac{17}{5}\right)\sqrt{50}\right)-\left(\left(8\right)\sqrt{8}+\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{18}\right)\right)\right)\\
&=&\left(-18-\left(\left(-\dfrac{175}{8}\right)\sqrt{2}+\left(\dfrac{71}{4}\right)\sqrt{2}-31\right)-\left(\left(35\right)\sqrt{2}+\left(24\right)\sqrt{2}+\dfrac{19}{7}+\dfrac{1}{2}\right)-\dfrac{44}{7}\right)-\left(\left(-\dfrac{53}{4}-\left(\left(0\right)\sqrt{2}\right)\right)-\left(-\dfrac{32}{5}+\left(-17\right)\sqrt{2}\right)-\left(\left(16\right)\sqrt{2}-\dfrac{1}{2}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(21\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{7}{2}+\left(-\dfrac{439}{8}\right)\sqrt{2}\right)-\left(-\dfrac{127}{20}+\left(\dfrac{227}{8}\right)\sqrt{2}\right)\\
&=&\dfrac{7}{2}+\left(-\dfrac{439}{8}\right)\sqrt{2}+\dfrac{127}{20}+\left(-\dfrac{227}{8}\right)\sqrt{2}\\
&=&\dfrac{197}{20}+\left(-\dfrac{333}{4}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-9\right)\sqrt{4}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{50}+\left(\dfrac{71}{8}\right)\sqrt{8}+\left(-\dfrac{31}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{8}+\left(8\right)\sqrt{18}+\dfrac{19}{7}+\left(\dfrac{1}{4}\right)\sqrt{4}\right)-\dfrac{44}{7}\right)\times\left(\left(\left(\left(-\dfrac{53}{8}\right)\sqrt{4}\right)-\left(\left(0\right)\sqrt{8}\right)\right)-\left(-\dfrac{32}{5}+\left(-\dfrac{17}{5}\right)\sqrt{50}\right)-\left(\left(8\right)\sqrt{8}+\left(-\dfrac{1}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{18}\right)\right)\right)\\
&=&\left(-18-\left(\left(-\dfrac{175}{8}\right)\sqrt{2}+\left(\dfrac{71}{4}\right)\sqrt{2}-31\right)-\left(\left(35\right)\sqrt{2}+\left(24\right)\sqrt{2}+\dfrac{19}{7}+\dfrac{1}{2}\right)-\dfrac{44}{7}\right)\times\left(\left(-\dfrac{53}{4}-\left(\left(0\right)\sqrt{2}\right)\right)-\left(-\dfrac{32}{5}+\left(-17\right)\sqrt{2}\right)-\left(\left(16\right)\sqrt{2}-\dfrac{1}{2}\right)-\left(\left(\left(-\dfrac{51}{8}\right)\sqrt{2}\right)-\left(\left(21\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{7}{2}+\left(-\dfrac{439}{8}\right)\sqrt{2}\right)\left(-\dfrac{127}{20}+\left(\dfrac{227}{8}\right)\sqrt{2}\right)\\
&=&-\dfrac{889}{40}+\left(\dfrac{71643}{160}\right)\sqrt{2}+\left(-\dfrac{99653}{64}\right)\sqrt{4}\\
&=&-\dfrac{501821}{160}+\left(\dfrac{71643}{160}\right)\sqrt{2}\\
\end{eqnarray*}