L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{8}\right)\right)-\left(\left(-4\right)\sqrt{4}\right)\) et \( Y=\left(-\dfrac{18}{5}\right)\sqrt{8}\) . 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{22}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{8}\right)\right)-\left(\left(-4\right)\sqrt{4}\right)\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}\right)+\dfrac{86}{7}-\left(\left(\dfrac{118}{9}\right)\sqrt{2}\right)\right)+8\right)+\left(\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}\right)+\dfrac{86}{7}-\left(\left(\dfrac{118}{9}\right)\sqrt{2}\right)\right)+8+\left(-\dfrac{36}{5}\right)\sqrt{2}\\
&=&\left(\dfrac{2599}{180}\right)\sqrt{2}+\dfrac{142}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{8}\right)\right)-\left(\left(-4\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{18}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}\right)+\dfrac{86}{7}-\left(\left(\dfrac{118}{9}\right)\sqrt{2}\right)\right)+8\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{779}{36}\right)\sqrt{2}+\dfrac{142}{7}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{779}{36}\right)\sqrt{2}+\dfrac{142}{7}+\left(\dfrac{36}{5}\right)\sqrt{2}\\
&=&\left(\dfrac{5191}{180}\right)\sqrt{2}+\dfrac{142}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{43}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{9}\right)\sqrt{8}\right)\right)-\left(\left(-4\right)\sqrt{4}\right)\right)\times\left(\left(-\dfrac{18}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\left(22\right)\sqrt{2}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}\right)+\dfrac{86}{7}-\left(\left(\dfrac{118}{9}\right)\sqrt{2}\right)\right)+8\right)\times\left(\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{779}{36}\right)\sqrt{2}+\dfrac{142}{7}\right)\left(\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{779}{5}\right)\sqrt{4}+\left(-\dfrac{5112}{35}\right)\sqrt{2}\\
&=&-\dfrac{1558}{5}+\left(-\dfrac{5112}{35}\right)\sqrt{2}\\
\end{eqnarray*}