L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{1}{5}\right)\sqrt{8}\) et \( Y=\left(\left(6\right)\sqrt{18}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{37}{9}\right)\sqrt{18}\right)+\left(-\dfrac{14}{3}\right)\sqrt{18}+\left(\dfrac{70}{9}\right)\sqrt{50}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(-\dfrac{6}{7}\right)\sqrt{18}+\left(-\dfrac{13}{4}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)+\left(\left(\left(6\right)\sqrt{18}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{37}{9}\right)\sqrt{18}\right)+\left(-\dfrac{14}{3}\right)\sqrt{18}+\left(\dfrac{70}{9}\right)\sqrt{50}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(-\dfrac{6}{7}\right)\sqrt{18}+\left(-\dfrac{13}{4}\right)\sqrt{50}\right)\\
&=&\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)+\left(\left(\left(18\right)\sqrt{2}\right)-\left(\left(-\dfrac{360}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{2}\right)+\left(-14\right)\sqrt{2}+\left(\dfrac{350}{9}\right)\sqrt{2}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(-\dfrac{18}{7}\right)\sqrt{2}+\left(-\dfrac{65}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{2}{5}\right)\sqrt{2}+\left(\left(18\right)\sqrt{2}\right)-\left(\left(-\dfrac{360}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{2}\right)+\left(-14\right)\sqrt{2}+\left(\dfrac{350}{9}\right)\sqrt{2}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(-\dfrac{18}{7}\right)\sqrt{2}+\left(-\dfrac{65}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{23243}{630}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)-\left(\left(\left(6\right)\sqrt{18}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{37}{9}\right)\sqrt{18}\right)+\left(-\dfrac{14}{3}\right)\sqrt{18}+\left(\dfrac{70}{9}\right)\sqrt{50}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(-\dfrac{6}{7}\right)\sqrt{18}+\left(-\dfrac{13}{4}\right)\sqrt{50}\right)\\
&=&\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)-\left(\left(\left(18\right)\sqrt{2}\right)-\left(\left(-\dfrac{360}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{2}\right)+\left(-14\right)\sqrt{2}+\left(\dfrac{350}{9}\right)\sqrt{2}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(-\dfrac{18}{7}\right)\sqrt{2}+\left(-\dfrac{65}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{4699}{126}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{2}{5}\right)\sqrt{2}+\left(-\dfrac{4699}{126}\right)\sqrt{2}\\
&=&\left(-\dfrac{23747}{630}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{1}{5}\right)\sqrt{8}\right)\times\left(\left(\left(6\right)\sqrt{18}\right)-\left(\left(-\dfrac{72}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{37}{9}\right)\sqrt{18}\right)+\left(-\dfrac{14}{3}\right)\sqrt{18}+\left(\dfrac{70}{9}\right)\sqrt{50}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(-\dfrac{6}{7}\right)\sqrt{18}+\left(-\dfrac{13}{4}\right)\sqrt{50}\right)\\
&=&\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)\times\left(\left(\left(18\right)\sqrt{2}\right)-\left(\left(-\dfrac{360}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{2}\right)+\left(-14\right)\sqrt{2}+\left(\dfrac{350}{9}\right)\sqrt{2}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(-\dfrac{18}{7}\right)\sqrt{2}+\left(-\dfrac{65}{4}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{2}{5}\right)\sqrt{2}\right)\left(\left(\dfrac{4699}{126}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{4699}{315}\right)\sqrt{4}\\
&=&-\dfrac{9398}{315}\\
\end{eqnarray*}