L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{19}{2}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{8}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{65}{7}\right)\sqrt{18}+\left(-\dfrac{61}{7}\right)\sqrt{8}\right)\) et \( Y=\left(\left(-9\right)\sqrt{18}+\dfrac{25}{3}+\left(\dfrac{11}{2}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{18}\right)-\dfrac{5}{2}-\left(\left(-2\right)\sqrt{4}\right)\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{50}\right)-\left(\left(-1\right)\sqrt{8}\right)\right)\) . 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(\dfrac{19}{2}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{8}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{65}{7}\right)\sqrt{18}+\left(-\dfrac{61}{7}\right)\sqrt{8}\right)\right)+\left(\left(\left(-9\right)\sqrt{18}+\dfrac{25}{3}+\left(\dfrac{11}{2}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{18}\right)-\dfrac{5}{2}-\left(\left(-2\right)\sqrt{4}\right)\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{50}\right)-\left(\left(-1\right)\sqrt{8}\right)\right)\right)\\
&=&\left(\left(19+\left(-15\right)\sqrt{2}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{195}{7}\right)\sqrt{2}+\left(-\dfrac{122}{7}\right)\sqrt{2}\right)\right)+\left(\left(\left(-27\right)\sqrt{2}+\dfrac{25}{3}+\left(11\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\dfrac{5}{2}+4\right)-\left(\dfrac{46}{7}-\left(\left(\dfrac{175}{6}\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)\right)\right)\\
&=&\left(19+\left(-15\right)\sqrt{2}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{195}{7}\right)\sqrt{2}+\left(-\dfrac{122}{7}\right)\sqrt{2}\right)+\left(\left(-27\right)\sqrt{2}+\dfrac{25}{3}+\left(11\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\dfrac{5}{2}+4\right)-\left(\dfrac{46}{7}-\left(\left(\dfrac{175}{6}\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)\right)\\
&=&\dfrac{953}{42}+\left(\dfrac{1741}{42}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{19}{2}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{8}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{65}{7}\right)\sqrt{18}+\left(-\dfrac{61}{7}\right)\sqrt{8}\right)\right)-\left(\left(\left(-9\right)\sqrt{18}+\dfrac{25}{3}+\left(\dfrac{11}{2}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{18}\right)-\dfrac{5}{2}-\left(\left(-2\right)\sqrt{4}\right)\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{50}\right)-\left(\left(-1\right)\sqrt{8}\right)\right)\right)\\
&=&\left(\left(19+\left(-15\right)\sqrt{2}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{195}{7}\right)\sqrt{2}+\left(-\dfrac{122}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(-27\right)\sqrt{2}+\dfrac{25}{3}+\left(11\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\dfrac{5}{2}+4\right)-\left(\dfrac{46}{7}-\left(\left(\dfrac{175}{6}\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{157}{7}+\left(\dfrac{212}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{67}{6}\right)\sqrt{2}+\dfrac{11}{42}\right)\\
&=&\dfrac{157}{7}+\left(\dfrac{212}{7}\right)\sqrt{2}+\left(-\dfrac{67}{6}\right)\sqrt{2}-\dfrac{11}{42}\\
&=&\dfrac{133}{6}+\left(\dfrac{803}{42}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{19}{2}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{8}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{65}{7}\right)\sqrt{18}+\left(-\dfrac{61}{7}\right)\sqrt{8}\right)\right)\times\left(\left(\left(-9\right)\sqrt{18}+\dfrac{25}{3}+\left(\dfrac{11}{2}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{18}\right)-\dfrac{5}{2}-\left(\left(-2\right)\sqrt{4}\right)\right)-\left(\left(\left(\dfrac{23}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{6}\right)\sqrt{50}\right)-\left(\left(-1\right)\sqrt{8}\right)\right)\right)\\
&=&\left(\left(19+\left(-15\right)\sqrt{2}\right)+\dfrac{24}{7}-\left(\left(-\dfrac{195}{7}\right)\sqrt{2}+\left(-\dfrac{122}{7}\right)\sqrt{2}\right)\right)\times\left(\left(\left(-27\right)\sqrt{2}+\dfrac{25}{3}+\left(11\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\dfrac{5}{2}+4\right)-\left(\dfrac{46}{7}-\left(\left(\dfrac{175}{6}\right)\sqrt{2}\right)-\left(\left(-2\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\dfrac{157}{7}+\left(\dfrac{212}{7}\right)\sqrt{2}\right)\left(\left(\dfrac{67}{6}\right)\sqrt{2}+\dfrac{11}{42}\right)\\
&=&\left(\dfrac{75965}{294}\right)\sqrt{2}+\dfrac{1727}{294}+\left(\dfrac{7102}{21}\right)\sqrt{4}\\
&=&\left(\dfrac{75965}{294}\right)\sqrt{2}+\dfrac{66861}{98}\\
\end{eqnarray*}