L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(-\dfrac{6}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{71}{4}\right)\sqrt{8}\right)-\dfrac{19}{4}-\left(\left(\dfrac{20}{3}\right)\sqrt{4}\right)\right)-\left(\left(5\right)\sqrt{50}\right)\) et \( Y=\left(\left(\dfrac{70}{3}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{18}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{8}\right)+\left(\left(0\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{50}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{6}\right)\sqrt{18}\right)\) . 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(\left(-\dfrac{6}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{71}{4}\right)\sqrt{8}\right)-\dfrac{19}{4}-\left(\left(\dfrac{20}{3}\right)\sqrt{4}\right)\right)-\left(\left(5\right)\sqrt{50}\right)\right)+\left(\left(\left(\dfrac{70}{3}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{18}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{8}\right)+\left(\left(0\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{50}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{6}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{2}\right)-\dfrac{19}{4}-\dfrac{40}{3}\right)-\left(\left(25\right)\sqrt{2}\right)\right)+\left(\left(\left(\dfrac{140}{3}\right)\sqrt{2}\right)-\left(\left(15\right)\sqrt{2}\right)-\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{2}\right)+0-37-\left(\left(-\dfrac{200}{3}\right)\sqrt{2}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{2}\right)-\dfrac{19}{4}-\dfrac{40}{3}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\left(\dfrac{140}{3}\right)\sqrt{2}\right)-\left(\left(15\right)\sqrt{2}\right)-\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{2}\right)+0-37-\left(\left(-\dfrac{200}{3}\right)\sqrt{2}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{995}{6}\right)\sqrt{2}-\dfrac{427}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{6}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{71}{4}\right)\sqrt{8}\right)-\dfrac{19}{4}-\left(\left(\dfrac{20}{3}\right)\sqrt{4}\right)\right)-\left(\left(5\right)\sqrt{50}\right)\right)-\left(\left(\left(\dfrac{70}{3}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{18}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{8}\right)+\left(\left(0\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{50}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{6}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{2}\right)-\dfrac{19}{4}-\dfrac{40}{3}\right)-\left(\left(25\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{140}{3}\right)\sqrt{2}\right)-\left(\left(15\right)\sqrt{2}\right)-\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{2}\right)+0-37-\left(\left(-\dfrac{200}{3}\right)\sqrt{2}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{217}{12}\right)-\left(\left(\dfrac{484}{3}\right)\sqrt{2}-\dfrac{35}{2}\right)\\
&=&\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{217}{12}+\left(-\dfrac{484}{3}\right)\sqrt{2}+\dfrac{35}{2}\\
&=&\left(-\dfrac{941}{6}\right)\sqrt{2}-\dfrac{7}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{6}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{71}{4}\right)\sqrt{8}\right)-\dfrac{19}{4}-\left(\left(\dfrac{20}{3}\right)\sqrt{4}\right)\right)-\left(\left(5\right)\sqrt{50}\right)\right)\times\left(\left(\left(\dfrac{70}{3}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{18}\right)-\left(\left(-\dfrac{71}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{8}\right)+\left(\left(0\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{40}{3}\right)\sqrt{50}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{6}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-6\right)\sqrt{2}\right)-\left(\left(-\dfrac{71}{2}\right)\sqrt{2}\right)-\dfrac{19}{4}-\dfrac{40}{3}\right)-\left(\left(25\right)\sqrt{2}\right)\right)\times\left(\left(\left(\dfrac{140}{3}\right)\sqrt{2}\right)-\left(\left(15\right)\sqrt{2}\right)-\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{21}{2}\right)\sqrt{2}\right)+0-37-\left(\left(-\dfrac{200}{3}\right)\sqrt{2}\right)+\dfrac{39}{2}-\left(\left(-\dfrac{5}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{217}{12}\right)\left(\left(\dfrac{484}{3}\right)\sqrt{2}-\dfrac{35}{2}\right)\\
&=&\left(726\right)\sqrt{4}+\left(-\dfrac{107863}{36}\right)\sqrt{2}+\dfrac{7595}{24}\\
&=&\dfrac{42443}{24}+\left(-\dfrac{107863}{36}\right)\sqrt{2}\\
\end{eqnarray*}