L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-1\right)\sqrt{8}\right)-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(8\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{4}\right)-3-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)\right)\) et \( Y=\left(\left(3\right)\sqrt{18}+\left(-\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)\right)\) . 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(\left(-1\right)\sqrt{8}\right)-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(8\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{4}\right)-3-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)\right)\right)+\left(\left(\left(3\right)\sqrt{18}+\left(-\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\left(\left(\left(-2\right)\sqrt{2}\right)-18\right)-16-\left(\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{9}\right)\sqrt{2}\right)+\dfrac{136}{7}-3-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(\left(12\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)\right)\right)+\left(\left(\left(9\right)\sqrt{2}+\left(-34\right)\sqrt{2}\right)-\left(\dfrac{23}{2}-\left(\left(11\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(\left(-2\right)\sqrt{2}\right)-18\right)-16-\left(\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{9}\right)\sqrt{2}\right)+\dfrac{136}{7}-3-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(\left(12\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)\right)+\left(\left(9\right)\sqrt{2}+\left(-34\right)\sqrt{2}\right)-\left(\dfrac{23}{2}-\left(\left(11\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{5354}{315}\right)\sqrt{2}-\dfrac{867}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-1\right)\sqrt{8}\right)-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(8\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{4}\right)-3-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)\right)\right)-\left(\left(\left(3\right)\sqrt{18}+\left(-\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\left(\left(\left(-2\right)\sqrt{2}\right)-18\right)-16-\left(\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{9}\right)\sqrt{2}\right)+\dfrac{136}{7}-3-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(\left(12\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)\right)\right)-\left(\left(\left(9\right)\sqrt{2}+\left(-34\right)\sqrt{2}\right)-\left(\dfrac{23}{2}-\left(\left(11\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-\dfrac{944}{315}\right)\sqrt{2}-\dfrac{353}{7}\right)-\left(\left(-14\right)\sqrt{2}-\dfrac{23}{2}\right)\\
&=&\left(-\dfrac{944}{315}\right)\sqrt{2}-\dfrac{353}{7}+\left(14\right)\sqrt{2}+\dfrac{23}{2}\\
&=&\left(\dfrac{3466}{315}\right)\sqrt{2}-\dfrac{545}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-1\right)\sqrt{8}\right)-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(8\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{4}\right)-3-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(\left(4\right)\sqrt{18}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)\right)\right)\times\left(\left(\left(3\right)\sqrt{18}+\left(-\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{23}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)\right)\right)\\
&=&\left(\left(\left(\left(-2\right)\sqrt{2}\right)-18\right)-16-\left(\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{9}\right)\sqrt{2}\right)+\dfrac{136}{7}-3-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(\left(12\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\left(9\right)\sqrt{2}+\left(-34\right)\sqrt{2}\right)-\left(\dfrac{23}{2}-\left(\left(11\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-\dfrac{944}{315}\right)\sqrt{2}-\dfrac{353}{7}\right)\left(\left(-14\right)\sqrt{2}-\dfrac{23}{2}\right)\\
&=&\left(\dfrac{1888}{45}\right)\sqrt{4}+\left(\dfrac{233246}{315}\right)\sqrt{2}+\dfrac{8119}{14}\\
&=&\dfrac{418219}{630}+\left(\dfrac{233246}{315}\right)\sqrt{2}\\
\end{eqnarray*}