L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(2\right)\sqrt{4}\right)+2-\left(\left(\dfrac{14}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{8}\right)\) et \( Y=\left(-7+\left(\dfrac{63}{4}\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{10}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)\right)\) . 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(2\right)\sqrt{4}\right)+2-\left(\left(\dfrac{14}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{8}\right)\right)+\left(\left(-7+\left(\dfrac{63}{4}\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{10}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)\right)\right)\\
&=&\left(\left(4+2-\left(\left(\dfrac{28}{5}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{2}\right)\right)+\left(\left(-7+\left(\dfrac{189}{4}\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{50}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(4+2-\left(\left(\dfrac{28}{5}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{2}\right)+\left(-7+\left(\dfrac{189}{4}\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{50}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)\right)\\
&=&\dfrac{33}{2}+\left(\dfrac{12547}{180}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(2\right)\sqrt{4}\right)+2-\left(\left(\dfrac{14}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{8}\right)\right)-\left(\left(-7+\left(\dfrac{63}{4}\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{10}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)\right)\right)\\
&=&\left(\left(4+2-\left(\left(\dfrac{28}{5}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{2}\right)\right)-\left(\left(-7+\left(\dfrac{189}{4}\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{50}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(6+\left(\dfrac{676}{15}\right)\sqrt{2}\right)-\left(\dfrac{21}{2}+\left(\dfrac{887}{36}\right)\sqrt{2}\right)\\
&=&6+\left(\dfrac{676}{15}\right)\sqrt{2}+-\dfrac{21}{2}+\left(-\dfrac{887}{36}\right)\sqrt{2}\\
&=&-\dfrac{9}{2}+\left(\dfrac{3677}{180}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(2\right)\sqrt{4}\right)+2-\left(\left(\dfrac{14}{5}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{8}\right)\right)\times\left(\left(-7+\left(\dfrac{63}{4}\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{10}{9}\right)\sqrt{50}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)\right)\right)\\
&=&\left(\left(4+2-\left(\left(\dfrac{28}{5}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{152}{3}\right)\sqrt{2}\right)\right)\times\left(\left(-7+\left(\dfrac{189}{4}\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}-\dfrac{35}{2}\right)-\left(\left(\left(-\dfrac{50}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)\right)\right)\\
&=&\left(6+\left(\dfrac{676}{15}\right)\sqrt{2}\right)\left(\dfrac{21}{2}+\left(\dfrac{887}{36}\right)\sqrt{2}\right)\\
&=&63+\left(\dfrac{18631}{30}\right)\sqrt{2}+\left(\dfrac{149903}{135}\right)\sqrt{4}\\
&=&\dfrac{308311}{135}+\left(\dfrac{18631}{30}\right)\sqrt{2}\\
\end{eqnarray*}