L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=-\dfrac{75}{4}+\left(\left(-5\right)\sqrt{63}\right)-\left(\left(-1\right)\sqrt{63}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{175}\right)+\left(-\dfrac{29}{3}\right)\sqrt{28}+\left(-\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{59}{8}\right)\sqrt{175}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{45}{2}\right)\sqrt{63}\) et \( Y=\left(\dfrac{25}{3}\right)\sqrt{28}+\left(\left(-\dfrac{1}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{49}\right)+\dfrac{11}{2}-\left(\left(-\dfrac{21}{4}\right)\sqrt{49}\right)+\left(2\right)\sqrt{63}+\left(-\dfrac{47}{9}\right)\sqrt{175}+\left(-\dfrac{32}{7}\right)\sqrt{28}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{75}{4}+\left(\left(-5\right)\sqrt{63}\right)-\left(\left(-1\right)\sqrt{63}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{175}\right)+\left(-\dfrac{29}{3}\right)\sqrt{28}+\left(-\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{59}{8}\right)\sqrt{175}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{45}{2}\right)\sqrt{63}\right)+\left(\left(\dfrac{25}{3}\right)\sqrt{28}+\left(\left(-\dfrac{1}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{49}\right)+\dfrac{11}{2}-\left(\left(-\dfrac{21}{4}\right)\sqrt{49}\right)+\left(2\right)\sqrt{63}+\left(-\dfrac{47}{9}\right)\sqrt{175}+\left(-\dfrac{32}{7}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{75}{4}+\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{295}{8}\right)\sqrt{7}\right)+\left(-\dfrac{58}{3}\right)\sqrt{7}+\left(-\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{295}{8}\right)\sqrt{7}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{135}{2}\right)\sqrt{7}\right)+\left(\left(\dfrac{50}{3}\right)\sqrt{7}+\left(\left(-\dfrac{1}{3}\right)\sqrt{7}\right)+\dfrac{532}{3}+\dfrac{11}{2}+\dfrac{147}{4}+\left(6\right)\sqrt{7}+\left(-\dfrac{235}{9}\right)\sqrt{7}+\left(-\dfrac{64}{7}\right)\sqrt{7}\right)\\
&=&-\dfrac{75}{4}+\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{295}{8}\right)\sqrt{7}\right)+\left(-\dfrac{58}{3}\right)\sqrt{7}+\left(-\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{295}{8}\right)\sqrt{7}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{135}{2}\right)\sqrt{7}+\left(\dfrac{50}{3}\right)\sqrt{7}+\left(\left(-\dfrac{1}{3}\right)\sqrt{7}\right)+\dfrac{532}{3}+\dfrac{11}{2}+\dfrac{147}{4}+\left(6\right)\sqrt{7}+\left(-\dfrac{235}{9}\right)\sqrt{7}+\left(-\dfrac{64}{7}\right)\sqrt{7}\\
&=&\dfrac{1213}{6}+\left(-\dfrac{4841}{126}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{75}{4}+\left(\left(-5\right)\sqrt{63}\right)-\left(\left(-1\right)\sqrt{63}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{175}\right)+\left(-\dfrac{29}{3}\right)\sqrt{28}+\left(-\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{59}{8}\right)\sqrt{175}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{45}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{28}+\left(\left(-\dfrac{1}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{49}\right)+\dfrac{11}{2}-\left(\left(-\dfrac{21}{4}\right)\sqrt{49}\right)+\left(2\right)\sqrt{63}+\left(-\dfrac{47}{9}\right)\sqrt{175}+\left(-\dfrac{32}{7}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{75}{4}+\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{295}{8}\right)\sqrt{7}\right)+\left(-\dfrac{58}{3}\right)\sqrt{7}+\left(-\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{295}{8}\right)\sqrt{7}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{135}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{7}+\left(\left(-\dfrac{1}{3}\right)\sqrt{7}\right)+\dfrac{532}{3}+\dfrac{11}{2}+\dfrac{147}{4}+\left(6\right)\sqrt{7}+\left(-\dfrac{235}{9}\right)\sqrt{7}+\left(-\dfrac{64}{7}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{209}{12}+\left(-\dfrac{51}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{814}{63}\right)\sqrt{7}+\dfrac{2635}{12}\right)\\
&=&-\dfrac{209}{12}+\left(-\dfrac{51}{2}\right)\sqrt{7}+\left(\dfrac{814}{63}\right)\sqrt{7}-\dfrac{2635}{12}\\
&=&-237+\left(-\dfrac{1585}{126}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{75}{4}+\left(\left(-5\right)\sqrt{63}\right)-\left(\left(-1\right)\sqrt{63}\right)-\left(\left(\dfrac{59}{8}\right)\sqrt{175}\right)+\left(-\dfrac{29}{3}\right)\sqrt{28}+\left(-\dfrac{37}{3}\right)\sqrt{175}+\left(\dfrac{59}{8}\right)\sqrt{175}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{45}{2}\right)\sqrt{63}\right)\times\left(\left(\dfrac{25}{3}\right)\sqrt{28}+\left(\left(-\dfrac{1}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{49}\right)+\dfrac{11}{2}-\left(\left(-\dfrac{21}{4}\right)\sqrt{49}\right)+\left(2\right)\sqrt{63}+\left(-\dfrac{47}{9}\right)\sqrt{175}+\left(-\dfrac{32}{7}\right)\sqrt{28}\right)\\
&=&\left(-\dfrac{75}{4}+\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-3\right)\sqrt{7}\right)-\left(\left(\dfrac{295}{8}\right)\sqrt{7}\right)+\left(-\dfrac{58}{3}\right)\sqrt{7}+\left(-\dfrac{185}{3}\right)\sqrt{7}+\left(\dfrac{295}{8}\right)\sqrt{7}+\dfrac{5}{6}+\dfrac{1}{2}+\left(\dfrac{135}{2}\right)\sqrt{7}\right)\times\left(\left(\dfrac{50}{3}\right)\sqrt{7}+\left(\left(-\dfrac{1}{3}\right)\sqrt{7}\right)+\dfrac{532}{3}+\dfrac{11}{2}+\dfrac{147}{4}+\left(6\right)\sqrt{7}+\left(-\dfrac{235}{9}\right)\sqrt{7}+\left(-\dfrac{64}{7}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{209}{12}+\left(-\dfrac{51}{2}\right)\sqrt{7}\right)\left(\left(-\dfrac{814}{63}\right)\sqrt{7}+\dfrac{2635}{12}\right)\\
&=&\left(-\dfrac{8126003}{1512}\right)\sqrt{7}-\dfrac{550715}{144}+\left(\dfrac{6919}{21}\right)\sqrt{49}\\
&=&\left(-\dfrac{8126003}{1512}\right)\sqrt{7}-\dfrac{218603}{144}\\
\end{eqnarray*}