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(\left(\dfrac{69}{5}\right)\sqrt{8}\right)-4-\left(\left(\dfrac{80}{7}\right)\sqrt{18}\right)+9-\left(\left(4\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{31}{8}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{18}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-\dfrac{71}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{8}\right)\right)$ et $ Y=\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(\dfrac{23}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{76}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{18}\right)\right)+\dfrac{17}{2}$ . 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(\left(\dfrac{69}{5}\right)\sqrt{8}\right)-4-\left(\left(\dfrac{80}{7}\right)\sqrt{18}\right)+9-\left(\left(4\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{31}{8}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{18}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-\dfrac{71}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{8}\right)\right)\right)+\left(\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(\dfrac{23}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{76}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{18}\right)\right)+\dfrac{17}{2}\right)\\ &=&\left(\left(\left(\left(\dfrac{138}{5}\right)\sqrt{2}\right)-4-\left(\left(\dfrac{240}{7}\right)\sqrt{2}\right)+9-\left(\left(20\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{2}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{2}\right)\right)\right)+\left(\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(23\right)\sqrt{2}\right)-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)+\dfrac{152}{7}-\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)\right)+\dfrac{17}{2}\right)\\ &=&\left(\left(\left(\dfrac{138}{5}\right)\sqrt{2}\right)-4-\left(\left(\dfrac{240}{7}\right)\sqrt{2}\right)+9-\left(\left(20\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{2}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{2}\right)\right)+\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(23\right)\sqrt{2}\right)-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)+\dfrac{152}{7}-\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)\right)+\dfrac{17}{2}\\ &=&\left(\dfrac{2881}{210}\right)\sqrt{2}-\dfrac{29}{168}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\left(\dfrac{69}{5}\right)\sqrt{8}\right)-4-\left(\left(\dfrac{80}{7}\right)\sqrt{18}\right)+9-\left(\left(4\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{31}{8}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{18}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-\dfrac{71}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{8}\right)\right)\right)-\left(\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(\dfrac{23}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{76}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{18}\right)\right)+\dfrac{17}{2}\right)\\ &=&\left(\left(\left(\left(\dfrac{138}{5}\right)\sqrt{2}\right)-4-\left(\left(\dfrac{240}{7}\right)\sqrt{2}\right)+9-\left(\left(20\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{2}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{2}\right)\right)\right)-\left(\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(23\right)\sqrt{2}\right)-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)+\dfrac{152}{7}-\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)\right)+\dfrac{17}{2}\right)\\ &=&\left(\left(-\dfrac{191}{140}\right)\sqrt{2}+\dfrac{14}{9}\right)-\left(-\dfrac{871}{504}+\left(\dfrac{181}{12}\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{191}{140}\right)\sqrt{2}+\dfrac{14}{9}+\dfrac{871}{504}+\left(-\dfrac{181}{12}\right)\sqrt{2}\\ &=&\left(-\dfrac{1727}{105}\right)\sqrt{2}+\dfrac{1655}{504}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\left(\dfrac{69}{5}\right)\sqrt{8}\right)-4-\left(\left(\dfrac{80}{7}\right)\sqrt{18}\right)+9-\left(\left(4\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{31}{8}\right)\sqrt{8}\right)-\left(\left(\left(0\right)\sqrt{50}\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{18}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-\dfrac{71}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{34}{7}\right)\sqrt{8}\right)\right)\right)\times\left(\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(\dfrac{23}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{76}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-2\right)\sqrt{18}\right)\right)+\dfrac{17}{2}\right)\\ &=&\left(\left(\left(\left(\dfrac{138}{5}\right)\sqrt{2}\right)-4-\left(\left(\dfrac{240}{7}\right)\sqrt{2}\right)+9-\left(\left(20\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{2}\right)-\left(\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{75}{2}\right)\sqrt{2}\right)-4+\dfrac{67}{9}\right)-\left(\left(\left(-71\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{68}{7}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\dfrac{37}{9}+\dfrac{59}{8}\right)-\left(\left(\left(23\right)\sqrt{2}\right)-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)+\dfrac{152}{7}-\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)\right)+\dfrac{17}{2}\right)\\ &=&\left(\left(-\dfrac{191}{140}\right)\sqrt{2}+\dfrac{14}{9}\right)\left(-\dfrac{871}{504}+\left(\dfrac{181}{12}\right)\sqrt{2}\right)\\ &=&\left(\dfrac{5465723}{211680}\right)\sqrt{2}+\left(-\dfrac{34571}{1680}\right)\sqrt{4}-\dfrac{871}{324}\\ &=&\left(\dfrac{5465723}{211680}\right)\sqrt{2}-\dfrac{994387}{22680}\\ \end{eqnarray*}