Ataraxy

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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.

Exercice

Soit $ X=\left(\left(\dfrac{18}{5}\right)\sqrt{18}\right)-\left(\left(4\right)\sqrt{8}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{50}-\dfrac{27}{7}+\left(\dfrac{18}{5}\right)\sqrt{18}+\left(-3\right)\sqrt{18}\right)$ et $ Y=-2+\left(-\dfrac{14}{3}\right)\sqrt{50}+\left(\dfrac{76}{5}\right)\sqrt{18}+\left(-1\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(\left(\dfrac{17}{7}\right)\sqrt{18}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{11}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{67}{3}\right)\sqrt{18}\right)+\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)+\left(-\dfrac{73}{3}\right)\sqrt{50}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\left(\left(\dfrac{18}{5}\right)\sqrt{18}\right)-\left(\left(4\right)\sqrt{8}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{50}-\dfrac{27}{7}+\left(\dfrac{18}{5}\right)\sqrt{18}+\left(-3\right)\sqrt{18}\right)\right)+\left(-2+\left(-\dfrac{14}{3}\right)\sqrt{50}+\left(\dfrac{76}{5}\right)\sqrt{18}+\left(-1\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(\left(\dfrac{17}{7}\right)\sqrt{18}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{11}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{67}{3}\right)\sqrt{18}\right)+\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)+\left(-\dfrac{73}{3}\right)\sqrt{50}\right)\\ &=&\left(\left(\left(\dfrac{54}{5}\right)\sqrt{2}\right)-\left(\left(8\right)\sqrt{2}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{2}-\dfrac{27}{7}+\left(\dfrac{54}{5}\right)\sqrt{2}+\left(-9\right)\sqrt{2}\right)\right)+\left(-2+\left(-\dfrac{70}{3}\right)\sqrt{2}+\left(\dfrac{228}{5}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+12+\left(\left(\dfrac{51}{7}\right)\sqrt{2}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}\right)-\left(\left(67\right)\sqrt{2}\right)+\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)+\left(-\dfrac{365}{3}\right)\sqrt{2}\right)\\ &=&\left(\left(\dfrac{54}{5}\right)\sqrt{2}\right)-\left(\left(8\right)\sqrt{2}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{2}-\dfrac{27}{7}+\left(\dfrac{54}{5}\right)\sqrt{2}+\left(-9\right)\sqrt{2}\right)-2+\left(-\dfrac{70}{3}\right)\sqrt{2}+\left(\dfrac{228}{5}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+12+\left(\left(\dfrac{51}{7}\right)\sqrt{2}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}\right)-\left(\left(67\right)\sqrt{2}\right)+\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)+\left(-\dfrac{365}{3}\right)\sqrt{2}\\ &=&\left(-\dfrac{22772}{105}\right)\sqrt{2}+\dfrac{79}{7}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{18}{5}\right)\sqrt{18}\right)-\left(\left(4\right)\sqrt{8}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{50}-\dfrac{27}{7}+\left(\dfrac{18}{5}\right)\sqrt{18}+\left(-3\right)\sqrt{18}\right)\right)-\left(-2+\left(-\dfrac{14}{3}\right)\sqrt{50}+\left(\dfrac{76}{5}\right)\sqrt{18}+\left(-1\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(\left(\dfrac{17}{7}\right)\sqrt{18}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{11}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{67}{3}\right)\sqrt{18}\right)+\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)+\left(-\dfrac{73}{3}\right)\sqrt{50}\right)\\ &=&\left(\left(\left(\dfrac{54}{5}\right)\sqrt{2}\right)-\left(\left(8\right)\sqrt{2}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{2}-\dfrac{27}{7}+\left(\dfrac{54}{5}\right)\sqrt{2}+\left(-9\right)\sqrt{2}\right)\right)-\left(-2+\left(-\dfrac{70}{3}\right)\sqrt{2}+\left(\dfrac{228}{5}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+12+\left(\left(\dfrac{51}{7}\right)\sqrt{2}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}\right)-\left(\left(67\right)\sqrt{2}\right)+\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)+\left(-\dfrac{365}{3}\right)\sqrt{2}\right)\\ &=&\left(\left(-\dfrac{355}{28}\right)\sqrt{2}+\dfrac{27}{7}\right)-\left(\dfrac{52}{7}+\left(-\dfrac{85763}{420}\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{355}{28}\right)\sqrt{2}+\dfrac{27}{7}+-\dfrac{52}{7}+\left(\dfrac{85763}{420}\right)\sqrt{2}\\ &=&\left(\dfrac{40219}{210}\right)\sqrt{2}-\dfrac{25}{7}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{18}{5}\right)\sqrt{18}\right)-\left(\left(4\right)\sqrt{8}\right)-\left(\left(\dfrac{64}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{50}-\dfrac{27}{7}+\left(\dfrac{18}{5}\right)\sqrt{18}+\left(-3\right)\sqrt{18}\right)\right)\times\left(-2+\left(-\dfrac{14}{3}\right)\sqrt{50}+\left(\dfrac{76}{5}\right)\sqrt{18}+\left(-1\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(\left(\dfrac{17}{7}\right)\sqrt{18}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{11}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{67}{3}\right)\sqrt{18}\right)+\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)+\left(-\dfrac{73}{3}\right)\sqrt{50}\right)\\ &=&\left(\left(\left(\dfrac{54}{5}\right)\sqrt{2}\right)-\left(\left(8\right)\sqrt{2}\right)-\left(\left(\dfrac{192}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{4}\right)\sqrt{2}-\dfrac{27}{7}+\left(\dfrac{54}{5}\right)\sqrt{2}+\left(-9\right)\sqrt{2}\right)\right)\times\left(-2+\left(-\dfrac{70}{3}\right)\sqrt{2}+\left(\dfrac{228}{5}\right)\sqrt{2}+\left(-2\right)\sqrt{2}+12+\left(\left(\dfrac{51}{7}\right)\sqrt{2}\right)-\dfrac{18}{7}-\left(\left(-\dfrac{33}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{112}{3}\right)\sqrt{2}\right)-\left(\left(67\right)\sqrt{2}\right)+\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)+\left(-\dfrac{365}{3}\right)\sqrt{2}\right)\\ &=&\left(\left(-\dfrac{355}{28}\right)\sqrt{2}+\dfrac{27}{7}\right)\left(\dfrac{52}{7}+\left(-\dfrac{85763}{420}\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{864167}{980}\right)\sqrt{2}+\left(\dfrac{6089173}{2352}\right)\sqrt{4}+\dfrac{1404}{49}\\ &=&\left(-\dfrac{864167}{980}\right)\sqrt{2}+\dfrac{6122869}{1176}\\ \end{eqnarray*}