L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(0\right)\sqrt{8}\right)-8\right)-\left(\left(\left(-\dfrac{61}{3}\right)\sqrt{18}\right)-5\right)-\left(\left(\left(-\dfrac{35}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{4}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{8}\right)\right)\) et \( Y=-\dfrac{5}{9}+\left(\dfrac{19}{4}\right)\sqrt{4}+\left(\dfrac{13}{2}\right)\sqrt{8}+\left(-\dfrac{5}{2}\right)\sqrt{18}\) . 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(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(0\right)\sqrt{8}\right)-8\right)-\left(\left(\left(-\dfrac{61}{3}\right)\sqrt{18}\right)-5\right)-\left(\left(\left(-\dfrac{35}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{4}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{8}\right)\right)\right)+\left(-\dfrac{5}{9}+\left(\dfrac{19}{4}\right)\sqrt{4}+\left(\dfrac{13}{2}\right)\sqrt{8}+\left(-\dfrac{5}{2}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(0\right)\sqrt{2}\right)-8\right)-\left(\left(\left(-61\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{35}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{4}\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)\right)+\left(-\dfrac{5}{9}+\dfrac{19}{2}+\left(13\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(0\right)\sqrt{2}\right)-8\right)-\left(\left(\left(-61\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{35}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{4}\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)-\dfrac{5}{9}+\dfrac{19}{2}+\left(13\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{71}{2}\right)\sqrt{2}+\dfrac{605}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(0\right)\sqrt{8}\right)-8\right)-\left(\left(\left(-\dfrac{61}{3}\right)\sqrt{18}\right)-5\right)-\left(\left(\left(-\dfrac{35}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{4}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{8}\right)\right)\right)-\left(-\dfrac{5}{9}+\left(\dfrac{19}{4}\right)\sqrt{4}+\left(\dfrac{13}{2}\right)\sqrt{8}+\left(-\dfrac{5}{2}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(0\right)\sqrt{2}\right)-8\right)-\left(\left(\left(-61\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{35}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{4}\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)\right)-\left(-\dfrac{5}{9}+\dfrac{19}{2}+\left(13\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-41\right)\sqrt{2}+\dfrac{74}{3}\right)-\left(\dfrac{161}{18}+\left(\dfrac{11}{2}\right)\sqrt{2}\right)\\
&=&\left(-41\right)\sqrt{2}+\dfrac{74}{3}+-\dfrac{161}{18}+\left(-\dfrac{11}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{93}{2}\right)\sqrt{2}+\dfrac{283}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{8}\right)\sqrt{8}\right)\right)-\left(\left(\left(0\right)\sqrt{8}\right)-8\right)-\left(\left(\left(-\dfrac{61}{3}\right)\sqrt{18}\right)-5\right)-\left(\left(\left(-\dfrac{35}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{53}{4}\right)\sqrt{50}\right)-\left(\left(-9\right)\sqrt{8}\right)\right)\right)\times\left(-\dfrac{5}{9}+\left(\dfrac{19}{4}\right)\sqrt{4}+\left(\dfrac{13}{2}\right)\sqrt{8}+\left(-\dfrac{5}{2}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{27}{4}\right)\sqrt{2}\right)\right)-\left(\left(\left(0\right)\sqrt{2}\right)-8\right)-\left(\left(\left(-61\right)\sqrt{2}\right)-5\right)-\left(-\dfrac{35}{3}-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{4}\right)\sqrt{2}\right)-\left(\left(-18\right)\sqrt{2}\right)\right)\right)\times\left(-\dfrac{5}{9}+\dfrac{19}{2}+\left(13\right)\sqrt{2}+\left(-\dfrac{15}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-41\right)\sqrt{2}+\dfrac{74}{3}\right)\left(\dfrac{161}{18}+\left(\dfrac{11}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{4159}{18}\right)\sqrt{2}+\left(-\dfrac{451}{2}\right)\sqrt{4}+\dfrac{5957}{27}\\
&=&\left(-\dfrac{4159}{18}\right)\sqrt{2}-\dfrac{6220}{27}\\
\end{eqnarray*}