L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{33}{4}\right)\sqrt{27}+\left(\dfrac{19}{2}\right)\sqrt{9}+\left(\left(-\dfrac{5}{9}\right)\sqrt{27}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{51}{8}\right)\sqrt{75}\right)\) et \( Y=\left(-\dfrac{51}{7}\right)\sqrt{9}+\left(\left(-\dfrac{49}{6}\right)\sqrt{9}\right)+\dfrac{49}{9}-\left(\left(-\dfrac{5}{4}\right)\sqrt{75}\right)+1+\left(-\dfrac{44}{7}\right)\sqrt{75}\) . 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{33}{4}\right)\sqrt{27}+\left(\dfrac{19}{2}\right)\sqrt{9}+\left(\left(-\dfrac{5}{9}\right)\sqrt{27}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{51}{8}\right)\sqrt{75}\right)\right)+\left(\left(-\dfrac{51}{7}\right)\sqrt{9}+\left(\left(-\dfrac{49}{6}\right)\sqrt{9}\right)+\dfrac{49}{9}-\left(\left(-\dfrac{5}{4}\right)\sqrt{75}\right)+1+\left(-\dfrac{44}{7}\right)\sqrt{75}\right)\\
&=&\left(\left(-\dfrac{99}{4}\right)\sqrt{3}+\dfrac{57}{2}+\left(\left(-\dfrac{5}{3}\right)\sqrt{3}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{255}{8}\right)\sqrt{3}\right)\right)+\left(-\dfrac{153}{7}-\dfrac{49}{2}+\dfrac{49}{9}-\left(\left(-\dfrac{25}{4}\right)\sqrt{3}\right)+1+\left(-\dfrac{220}{7}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{99}{4}\right)\sqrt{3}+\dfrac{57}{2}+\left(\left(-\dfrac{5}{3}\right)\sqrt{3}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{255}{8}\right)\sqrt{3}\right)-\dfrac{153}{7}-\dfrac{49}{2}+\dfrac{49}{9}-\left(\left(-\dfrac{25}{4}\right)\sqrt{3}\right)+1+\left(-\dfrac{220}{7}\right)\sqrt{3}\\
&=&\left(-\dfrac{3313}{168}\right)\sqrt{3}-\dfrac{2719}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{33}{4}\right)\sqrt{27}+\left(\dfrac{19}{2}\right)\sqrt{9}+\left(\left(-\dfrac{5}{9}\right)\sqrt{27}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{51}{8}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{51}{7}\right)\sqrt{9}+\left(\left(-\dfrac{49}{6}\right)\sqrt{9}\right)+\dfrac{49}{9}-\left(\left(-\dfrac{5}{4}\right)\sqrt{75}\right)+1+\left(-\dfrac{44}{7}\right)\sqrt{75}\right)\\
&=&\left(\left(-\dfrac{99}{4}\right)\sqrt{3}+\dfrac{57}{2}+\left(\left(-\dfrac{5}{3}\right)\sqrt{3}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{255}{8}\right)\sqrt{3}\right)\right)-\left(-\dfrac{153}{7}-\dfrac{49}{2}+\dfrac{49}{9}-\left(\left(-\dfrac{25}{4}\right)\sqrt{3}\right)+1+\left(-\dfrac{220}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{131}{24}\right)\sqrt{3}+\dfrac{55}{3}\right)-\left(-\dfrac{5029}{126}+\left(-\dfrac{705}{28}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{131}{24}\right)\sqrt{3}+\dfrac{55}{3}+\dfrac{5029}{126}+\left(\dfrac{705}{28}\right)\sqrt{3}\\
&=&\left(\dfrac{5147}{168}\right)\sqrt{3}+\dfrac{7339}{126}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{33}{4}\right)\sqrt{27}+\left(\dfrac{19}{2}\right)\sqrt{9}+\left(\left(-\dfrac{5}{9}\right)\sqrt{27}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{51}{8}\right)\sqrt{75}\right)\right)\times\left(\left(-\dfrac{51}{7}\right)\sqrt{9}+\left(\left(-\dfrac{49}{6}\right)\sqrt{9}\right)+\dfrac{49}{9}-\left(\left(-\dfrac{5}{4}\right)\sqrt{75}\right)+1+\left(-\dfrac{44}{7}\right)\sqrt{75}\right)\\
&=&\left(\left(-\dfrac{99}{4}\right)\sqrt{3}+\dfrac{57}{2}+\left(\left(-\dfrac{5}{3}\right)\sqrt{3}\right)-\dfrac{61}{6}-\left(\left(-\dfrac{255}{8}\right)\sqrt{3}\right)\right)\times\left(-\dfrac{153}{7}-\dfrac{49}{2}+\dfrac{49}{9}-\left(\left(-\dfrac{25}{4}\right)\sqrt{3}\right)+1+\left(-\dfrac{220}{7}\right)\sqrt{3}\right)\\
&=&\left(\left(\dfrac{131}{24}\right)\sqrt{3}+\dfrac{55}{3}\right)\left(-\dfrac{5029}{126}+\left(-\dfrac{705}{28}\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{2054699}{3024}\right)\sqrt{3}+\left(-\dfrac{30785}{224}\right)\sqrt{9}-\dfrac{276595}{378}\\
&=&\left(-\dfrac{2054699}{3024}\right)\sqrt{3}-\dfrac{6919105}{6048}\\
\end{eqnarray*}