L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{71}{3}\right)\sqrt{4}\) et \( Y=\left(5\right)\sqrt{4}+\left(-\dfrac{37}{9}\right)\sqrt{50}+\dfrac{19}{2}+\left(3\right)\sqrt{50}+\left(\dfrac{57}{5}\right)\sqrt{18}+\left(6\right)\sqrt{18}+\left(\dfrac{19}{3}\right)\sqrt{18}-\dfrac{16}{3}+\left(-\dfrac{55}{2}\right)\sqrt{8}-\dfrac{29}{6}\) . 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{71}{3}\right)\sqrt{4}\right)+\left(\left(5\right)\sqrt{4}+\left(-\dfrac{37}{9}\right)\sqrt{50}+\dfrac{19}{2}+\left(3\right)\sqrt{50}+\left(\dfrac{57}{5}\right)\sqrt{18}+\left(6\right)\sqrt{18}+\left(\dfrac{19}{3}\right)\sqrt{18}-\dfrac{16}{3}+\left(-\dfrac{55}{2}\right)\sqrt{8}-\dfrac{29}{6}\right)\\
&=&\left(\dfrac{142}{3}\right)+\left(10+\left(-\dfrac{185}{9}\right)\sqrt{2}+\dfrac{19}{2}+\left(15\right)\sqrt{2}+\left(\dfrac{171}{5}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(19\right)\sqrt{2}-\dfrac{16}{3}+\left(-55\right)\sqrt{2}-\dfrac{29}{6}\right)\\
&=&\dfrac{142}{3}+10+\left(-\dfrac{185}{9}\right)\sqrt{2}+\dfrac{19}{2}+\left(15\right)\sqrt{2}+\left(\dfrac{171}{5}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(19\right)\sqrt{2}-\dfrac{16}{3}+\left(-55\right)\sqrt{2}-\dfrac{29}{6}\\
&=&\dfrac{170}{3}+\left(\dfrac{479}{45}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{71}{3}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{4}+\left(-\dfrac{37}{9}\right)\sqrt{50}+\dfrac{19}{2}+\left(3\right)\sqrt{50}+\left(\dfrac{57}{5}\right)\sqrt{18}+\left(6\right)\sqrt{18}+\left(\dfrac{19}{3}\right)\sqrt{18}-\dfrac{16}{3}+\left(-\dfrac{55}{2}\right)\sqrt{8}-\dfrac{29}{6}\right)\\
&=&\left(\dfrac{142}{3}\right)-\left(10+\left(-\dfrac{185}{9}\right)\sqrt{2}+\dfrac{19}{2}+\left(15\right)\sqrt{2}+\left(\dfrac{171}{5}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(19\right)\sqrt{2}-\dfrac{16}{3}+\left(-55\right)\sqrt{2}-\dfrac{29}{6}\right)\\
&=&\left(\dfrac{142}{3}\right)-\left(\dfrac{28}{3}+\left(\dfrac{479}{45}\right)\sqrt{2}\right)\\
&=&\dfrac{142}{3}+-\dfrac{28}{3}+\left(-\dfrac{479}{45}\right)\sqrt{2}\\
&=&38+\left(-\dfrac{479}{45}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{71}{3}\right)\sqrt{4}\right)\times\left(\left(5\right)\sqrt{4}+\left(-\dfrac{37}{9}\right)\sqrt{50}+\dfrac{19}{2}+\left(3\right)\sqrt{50}+\left(\dfrac{57}{5}\right)\sqrt{18}+\left(6\right)\sqrt{18}+\left(\dfrac{19}{3}\right)\sqrt{18}-\dfrac{16}{3}+\left(-\dfrac{55}{2}\right)\sqrt{8}-\dfrac{29}{6}\right)\\
&=&\left(\dfrac{142}{3}\right)\times\left(10+\left(-\dfrac{185}{9}\right)\sqrt{2}+\dfrac{19}{2}+\left(15\right)\sqrt{2}+\left(\dfrac{171}{5}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(19\right)\sqrt{2}-\dfrac{16}{3}+\left(-55\right)\sqrt{2}-\dfrac{29}{6}\right)\\
&=&\left(\dfrac{142}{3}\right)\left(\dfrac{28}{3}+\left(\dfrac{479}{45}\right)\sqrt{2}\right)\\
&=&\dfrac{3976}{9}+\left(\dfrac{68018}{135}\right)\sqrt{2}\\
&=&\dfrac{3976}{9}+\left(\dfrac{68018}{135}\right)\sqrt{2}\\
\end{eqnarray*}