L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{65}{8}\right)\sqrt{50}+\left(\dfrac{31}{2}\right)\sqrt{4}+\dfrac{74}{3}+\left(\dfrac{31}{2}\right)\sqrt{4}\) et \( Y=\left(\dfrac{3}{7}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{65}{8}\right)\sqrt{50}+\left(\dfrac{31}{2}\right)\sqrt{4}+\dfrac{74}{3}+\left(\dfrac{31}{2}\right)\sqrt{4}\right)+\left(\left(\dfrac{3}{7}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{325}{8}\right)\sqrt{2}+31+\dfrac{74}{3}+31\right)+\left(\left(\dfrac{9}{7}\right)\sqrt{2}\right)\\
&=&\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{325}{8}\right)\sqrt{2}+31+\dfrac{74}{3}+31+\left(\dfrac{9}{7}\right)\sqrt{2}\\
&=&\dfrac{2215}{21}+\left(\dfrac{2347}{56}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{65}{8}\right)\sqrt{50}+\left(\dfrac{31}{2}\right)\sqrt{4}+\dfrac{74}{3}+\left(\dfrac{31}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{3}{7}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{325}{8}\right)\sqrt{2}+31+\dfrac{74}{3}+31\right)-\left(\left(\dfrac{9}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2215}{21}+\left(\dfrac{325}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{9}{7}\right)\sqrt{2}\right)\\
&=&\dfrac{2215}{21}+\left(\dfrac{325}{8}\right)\sqrt{2}+\left(-\dfrac{9}{7}\right)\sqrt{2}\\
&=&\dfrac{2215}{21}+\left(\dfrac{2203}{56}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{65}{8}\right)\sqrt{50}+\left(\dfrac{31}{2}\right)\sqrt{4}+\dfrac{74}{3}+\left(\dfrac{31}{2}\right)\sqrt{4}\right)\times\left(\left(\dfrac{3}{7}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{59}{3}-\dfrac{6}{7}+\left(\dfrac{325}{8}\right)\sqrt{2}+31+\dfrac{74}{3}+31\right)\times\left(\left(\dfrac{9}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2215}{21}+\left(\dfrac{325}{8}\right)\sqrt{2}\right)\left(\left(\dfrac{9}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{6645}{49}\right)\sqrt{2}+\left(\dfrac{2925}{56}\right)\sqrt{4}\\
&=&\left(\dfrac{6645}{49}\right)\sqrt{2}+\dfrac{2925}{28}\\
\end{eqnarray*}