L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{76}{3}-\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\left(-\dfrac{2}{7}\right)\sqrt{18}+\left(\dfrac{21}{2}\right)\sqrt{18}\) et \( Y=\left(-7\right)\sqrt{50}+\left(\left(\dfrac{29}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{8}\right)+\left(\dfrac{20}{3}\right)\sqrt{8}+\left(\dfrac{49}{6}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{18}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{76}{3}-\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\left(-\dfrac{2}{7}\right)\sqrt{18}+\left(\dfrac{21}{2}\right)\sqrt{18}\right)+\left(\left(-7\right)\sqrt{50}+\left(\left(\dfrac{29}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{8}\right)+\left(\dfrac{20}{3}\right)\sqrt{8}+\left(\dfrac{49}{6}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{76}{3}-\left(\left(4\right)\sqrt{2}\right)+\left(-\dfrac{6}{7}\right)\sqrt{2}+\left(\dfrac{63}{2}\right)\sqrt{2}\right)+\left(\left(-35\right)\sqrt{2}+\left(\left(29\right)\sqrt{2}\right)-\left(\left(\dfrac{365}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{2}\right)+\left(\dfrac{40}{3}\right)\sqrt{2}+\left(\dfrac{245}{6}\right)\sqrt{2}+\left(-\dfrac{75}{8}\right)\sqrt{2}\right)\\
&=&\dfrac{76}{3}-\left(\left(4\right)\sqrt{2}\right)+\left(-\dfrac{6}{7}\right)\sqrt{2}+\left(\dfrac{63}{2}\right)\sqrt{2}+\left(-35\right)\sqrt{2}+\left(\left(29\right)\sqrt{2}\right)-\left(\left(\dfrac{365}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{2}\right)+\left(\dfrac{40}{3}\right)\sqrt{2}+\left(\dfrac{245}{6}\right)\sqrt{2}+\left(-\dfrac{75}{8}\right)\sqrt{2}\\
&=&\dfrac{76}{3}+\left(\dfrac{433}{24}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{76}{3}-\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\left(-\dfrac{2}{7}\right)\sqrt{18}+\left(\dfrac{21}{2}\right)\sqrt{18}\right)-\left(\left(-7\right)\sqrt{50}+\left(\left(\dfrac{29}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{8}\right)+\left(\dfrac{20}{3}\right)\sqrt{8}+\left(\dfrac{49}{6}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{76}{3}-\left(\left(4\right)\sqrt{2}\right)+\left(-\dfrac{6}{7}\right)\sqrt{2}+\left(\dfrac{63}{2}\right)\sqrt{2}\right)-\left(\left(-35\right)\sqrt{2}+\left(\left(29\right)\sqrt{2}\right)-\left(\left(\dfrac{365}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{2}\right)+\left(\dfrac{40}{3}\right)\sqrt{2}+\left(\dfrac{245}{6}\right)\sqrt{2}+\left(-\dfrac{75}{8}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{76}{3}+\left(\dfrac{373}{14}\right)\sqrt{2}\right)-\left(\left(-\dfrac{1445}{168}\right)\sqrt{2}\right)\\
&=&\dfrac{76}{3}+\left(\dfrac{373}{14}\right)\sqrt{2}+\left(\dfrac{1445}{168}\right)\sqrt{2}\\
&=&\dfrac{76}{3}+\left(\dfrac{5921}{168}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{76}{3}-\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\left(-\dfrac{2}{7}\right)\sqrt{18}+\left(\dfrac{21}{2}\right)\sqrt{18}\right)\times\left(\left(-7\right)\sqrt{50}+\left(\left(\dfrac{29}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{73}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{8}\right)+\left(\dfrac{20}{3}\right)\sqrt{8}+\left(\dfrac{49}{6}\right)\sqrt{50}+\left(-\dfrac{25}{8}\right)\sqrt{18}\right)\\
&=&\left(\dfrac{76}{3}-\left(\left(4\right)\sqrt{2}\right)+\left(-\dfrac{6}{7}\right)\sqrt{2}+\left(\dfrac{63}{2}\right)\sqrt{2}\right)\times\left(\left(-35\right)\sqrt{2}+\left(\left(29\right)\sqrt{2}\right)-\left(\left(\dfrac{365}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{19}{4}\right)\sqrt{2}\right)+\left(\dfrac{40}{3}\right)\sqrt{2}+\left(\dfrac{245}{6}\right)\sqrt{2}+\left(-\dfrac{75}{8}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{76}{3}+\left(\dfrac{373}{14}\right)\sqrt{2}\right)\left(\left(-\dfrac{1445}{168}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{27455}{126}\right)\sqrt{2}+\left(-\dfrac{538985}{2352}\right)\sqrt{4}\\
&=&\left(-\dfrac{27455}{126}\right)\sqrt{2}-\dfrac{538985}{1176}\\
\end{eqnarray*}