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=-\dfrac{24}{5}$ et $ Y=\left(\dfrac{63}{4}\right)\sqrt{8}+\left(-\dfrac{17}{7}\right)\sqrt{8}+\left(-\dfrac{23}{7}\right)\sqrt{8}+\dfrac{29}{2}+\left(\left(\dfrac{1}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{18}\right)+\dfrac{25}{2}-\left(\left(8\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{18}\right)+\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\dfrac{2}{7}-\left(\left(\dfrac{54}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{8}\right)-\dfrac{29}{2}-\dfrac{47}{7}+\left(\dfrac{17}{2}\right)\sqrt{4}-1+\left(-\dfrac{19}{4}\right)\sqrt{50}+\left(0\right)\sqrt{4}+\left(\left(-9\right)\sqrt{50}\right)-\left(\left(\dfrac{34}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{77}{6}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{4}\right)+\dfrac{59}{9}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(-\dfrac{24}{5}\right)+\left(\left(\dfrac{63}{4}\right)\sqrt{8}+\left(-\dfrac{17}{7}\right)\sqrt{8}+\left(-\dfrac{23}{7}\right)\sqrt{8}+\dfrac{29}{2}+\left(\left(\dfrac{1}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{18}\right)+\dfrac{25}{2}-\left(\left(8\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{18}\right)+\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\dfrac{2}{7}-\left(\left(\dfrac{54}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{8}\right)-\dfrac{29}{2}-\dfrac{47}{7}+\left(\dfrac{17}{2}\right)\sqrt{4}-1+\left(-\dfrac{19}{4}\right)\sqrt{50}+\left(0\right)\sqrt{4}+\left(\left(-9\right)\sqrt{50}\right)-\left(\left(\dfrac{34}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{77}{6}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{4}\right)+\dfrac{59}{9}\right)\\ &=&\left(-\dfrac{24}{5}\right)+\left(\left(\dfrac{63}{2}\right)\sqrt{2}+\left(-\dfrac{34}{7}\right)\sqrt{2}+\left(-\dfrac{46}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(\left(\dfrac{5}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{2}\right)+\dfrac{25}{2}-\left(\left(40\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)+\left(\left(4\right)\sqrt{2}\right)+\dfrac{2}{7}-\left(\left(\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{2}\right)-\dfrac{29}{2}-\dfrac{47}{7}+17-1+\left(-\dfrac{95}{4}\right)\sqrt{2}+0+\left(\left(-45\right)\sqrt{2}\right)-\left(\left(\dfrac{68}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{2}\right)-10+\dfrac{59}{9}\right)\\ &=&-\dfrac{24}{5}+\left(\dfrac{63}{2}\right)\sqrt{2}+\left(-\dfrac{34}{7}\right)\sqrt{2}+\left(-\dfrac{46}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(\left(\dfrac{5}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{2}\right)+\dfrac{25}{2}-\left(\left(40\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)+\left(\left(4\right)\sqrt{2}\right)+\dfrac{2}{7}-\left(\left(\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{2}\right)-\dfrac{29}{2}-\dfrac{47}{7}+17-1+\left(-\dfrac{95}{4}\right)\sqrt{2}+0+\left(\left(-45\right)\sqrt{2}\right)-\left(\left(\dfrac{68}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{2}\right)-10+\dfrac{59}{9}\\ &=&\dfrac{8711}{630}+\left(-\dfrac{2859}{28}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(-\dfrac{24}{5}\right)-\left(\left(\dfrac{63}{4}\right)\sqrt{8}+\left(-\dfrac{17}{7}\right)\sqrt{8}+\left(-\dfrac{23}{7}\right)\sqrt{8}+\dfrac{29}{2}+\left(\left(\dfrac{1}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{18}\right)+\dfrac{25}{2}-\left(\left(8\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{18}\right)+\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\dfrac{2}{7}-\left(\left(\dfrac{54}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{8}\right)-\dfrac{29}{2}-\dfrac{47}{7}+\left(\dfrac{17}{2}\right)\sqrt{4}-1+\left(-\dfrac{19}{4}\right)\sqrt{50}+\left(0\right)\sqrt{4}+\left(\left(-9\right)\sqrt{50}\right)-\left(\left(\dfrac{34}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{77}{6}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{4}\right)+\dfrac{59}{9}\right)\\ &=&\left(-\dfrac{24}{5}\right)-\left(\left(\dfrac{63}{2}\right)\sqrt{2}+\left(-\dfrac{34}{7}\right)\sqrt{2}+\left(-\dfrac{46}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(\left(\dfrac{5}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{2}\right)+\dfrac{25}{2}-\left(\left(40\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)+\left(\left(4\right)\sqrt{2}\right)+\dfrac{2}{7}-\left(\left(\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{2}\right)-\dfrac{29}{2}-\dfrac{47}{7}+17-1+\left(-\dfrac{95}{4}\right)\sqrt{2}+0+\left(\left(-45\right)\sqrt{2}\right)-\left(\left(\dfrac{68}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{2}\right)-10+\dfrac{59}{9}\right)\\ &=&\left(-\dfrac{24}{5}\right)-\left(\left(-\dfrac{2859}{28}\right)\sqrt{2}+\dfrac{2347}{126}\right)\\ &=&-\dfrac{24}{5}+\left(\dfrac{2859}{28}\right)\sqrt{2}-\dfrac{2347}{126}\\ &=&-\dfrac{14759}{630}+\left(\dfrac{2859}{28}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(-\dfrac{24}{5}\right)\times\left(\left(\dfrac{63}{4}\right)\sqrt{8}+\left(-\dfrac{17}{7}\right)\sqrt{8}+\left(-\dfrac{23}{7}\right)\sqrt{8}+\dfrac{29}{2}+\left(\left(\dfrac{1}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{18}\right)+\dfrac{25}{2}-\left(\left(8\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{3}\right)\sqrt{18}\right)+\left(\left(\dfrac{4}{3}\right)\sqrt{18}\right)+\dfrac{2}{7}-\left(\left(\dfrac{54}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{8}\right)-\dfrac{29}{2}-\dfrac{47}{7}+\left(\dfrac{17}{2}\right)\sqrt{4}-1+\left(-\dfrac{19}{4}\right)\sqrt{50}+\left(0\right)\sqrt{4}+\left(\left(-9\right)\sqrt{50}\right)-\left(\left(\dfrac{34}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{77}{6}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{4}\right)+\dfrac{59}{9}\right)\\ &=&\left(-\dfrac{24}{5}\right)\times\left(\left(\dfrac{63}{2}\right)\sqrt{2}+\left(-\dfrac{34}{7}\right)\sqrt{2}+\left(-\dfrac{46}{7}\right)\sqrt{2}+\dfrac{29}{2}+\left(\left(\dfrac{5}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{2}\right)+\dfrac{25}{2}-\left(\left(40\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)+\left(\left(4\right)\sqrt{2}\right)+\dfrac{2}{7}-\left(\left(\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{57}{2}\right)\sqrt{2}\right)-\dfrac{29}{2}-\dfrac{47}{7}+17-1+\left(-\dfrac{95}{4}\right)\sqrt{2}+0+\left(\left(-45\right)\sqrt{2}\right)-\left(\left(\dfrac{68}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{77}{3}\right)\sqrt{2}\right)-10+\dfrac{59}{9}\right)\\ &=&\left(-\dfrac{24}{5}\right)\left(\left(-\dfrac{2859}{28}\right)\sqrt{2}+\dfrac{2347}{126}\right)\\ &=&\left(\dfrac{17154}{35}\right)\sqrt{2}-\dfrac{9388}{105}\\ &=&\left(\dfrac{17154}{35}\right)\sqrt{2}-\dfrac{9388}{105}\\ \end{eqnarray*}