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=-9+\left(-\dfrac{33}{4}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}+\left(\dfrac{60}{7}\right)\sqrt{28}+\left(\dfrac{53}{3}\right)\sqrt{28}+\left(-\dfrac{7}{8}\right)\sqrt{49}+\left(\dfrac{65}{6}\right)\sqrt{63}+\left(\left(-\dfrac{34}{5}\right)\sqrt{63}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{22}{7}\right)\sqrt{63}\right)+\left(\left(\dfrac{75}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{175}\right)-\left(\left(5\right)\sqrt{49}\right)-\left(\left(-5\right)\sqrt{49}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{63}\right)$ et $ Y=\dfrac{53}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{49}+\left(\dfrac{81}{5}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{63}\right)-\left(\left(7\right)\sqrt{28}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{175}+\left(-\dfrac{19}{3}\right)\sqrt{63}\right)$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(-9+\left(-\dfrac{33}{4}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}+\left(\dfrac{60}{7}\right)\sqrt{28}+\left(\dfrac{53}{3}\right)\sqrt{28}+\left(-\dfrac{7}{8}\right)\sqrt{49}+\left(\dfrac{65}{6}\right)\sqrt{63}+\left(\left(-\dfrac{34}{5}\right)\sqrt{63}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{22}{7}\right)\sqrt{63}\right)+\left(\left(\dfrac{75}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{175}\right)-\left(\left(5\right)\sqrt{49}\right)-\left(\left(-5\right)\sqrt{49}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{63}\right)\right)+\left(\dfrac{53}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{49}+\left(\dfrac{81}{5}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{63}\right)-\left(\left(7\right)\sqrt{28}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{175}+\left(-\dfrac{19}{3}\right)\sqrt{63}\right)\right)\\ &=&\left(-9+\left(-\dfrac{165}{4}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}+\left(\dfrac{120}{7}\right)\sqrt{7}+\left(\dfrac{106}{3}\right)\sqrt{7}-\dfrac{49}{8}+\left(\dfrac{65}{2}\right)\sqrt{7}+\left(\left(-\dfrac{102}{5}\right)\sqrt{7}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{66}{7}\right)\sqrt{7}\right)+\left(\left(\dfrac{150}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{2}\right)\sqrt{7}\right)-35+35-\left(\left(\dfrac{52}{3}\right)\sqrt{7}\right)\right)+\left(\dfrac{53}{5}-\left(\dfrac{567}{5}+\dfrac{567}{5}+\left(-\dfrac{154}{5}\right)\sqrt{7}\right)-\left(\left(11\right)\sqrt{7}\right)-\left(\left(14\right)\sqrt{7}\right)-\left(\left(-19\right)\sqrt{7}+\left(-19\right)\sqrt{7}\right)\right)\\ &=&-9+\left(-\dfrac{165}{4}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}+\left(\dfrac{120}{7}\right)\sqrt{7}+\left(\dfrac{106}{3}\right)\sqrt{7}-\dfrac{49}{8}+\left(\dfrac{65}{2}\right)\sqrt{7}+\left(\left(-\dfrac{102}{5}\right)\sqrt{7}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{66}{7}\right)\sqrt{7}\right)+\left(\left(\dfrac{150}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{2}\right)\sqrt{7}\right)-35+35-\left(\left(\dfrac{52}{3}\right)\sqrt{7}\right)+\dfrac{53}{5}-\left(\dfrac{567}{5}+\dfrac{567}{5}+\left(-\dfrac{154}{5}\right)\sqrt{7}\right)-\left(\left(11\right)\sqrt{7}\right)-\left(\left(14\right)\sqrt{7}\right)-\left(\left(-19\right)\sqrt{7}+\left(-19\right)\sqrt{7}\right)\\ &=&-\dfrac{27679}{120}+\left(\dfrac{769}{60}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(-9+\left(-\dfrac{33}{4}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}+\left(\dfrac{60}{7}\right)\sqrt{28}+\left(\dfrac{53}{3}\right)\sqrt{28}+\left(-\dfrac{7}{8}\right)\sqrt{49}+\left(\dfrac{65}{6}\right)\sqrt{63}+\left(\left(-\dfrac{34}{5}\right)\sqrt{63}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{22}{7}\right)\sqrt{63}\right)+\left(\left(\dfrac{75}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{175}\right)-\left(\left(5\right)\sqrt{49}\right)-\left(\left(-5\right)\sqrt{49}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{63}\right)\right)-\left(\dfrac{53}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{49}+\left(\dfrac{81}{5}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{63}\right)-\left(\left(7\right)\sqrt{28}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{175}+\left(-\dfrac{19}{3}\right)\sqrt{63}\right)\right)\\ &=&\left(-9+\left(-\dfrac{165}{4}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}+\left(\dfrac{120}{7}\right)\sqrt{7}+\left(\dfrac{106}{3}\right)\sqrt{7}-\dfrac{49}{8}+\left(\dfrac{65}{2}\right)\sqrt{7}+\left(\left(-\dfrac{102}{5}\right)\sqrt{7}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{66}{7}\right)\sqrt{7}\right)+\left(\left(\dfrac{150}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{2}\right)\sqrt{7}\right)-35+35-\left(\left(\dfrac{52}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{53}{5}-\left(\dfrac{567}{5}+\dfrac{567}{5}+\left(-\dfrac{154}{5}\right)\sqrt{7}\right)-\left(\left(11\right)\sqrt{7}\right)-\left(\left(14\right)\sqrt{7}\right)-\left(\left(-19\right)\sqrt{7}+\left(-19\right)\sqrt{7}\right)\right)\\ &=&\left(-\dfrac{347}{24}+\left(-\dfrac{1859}{60}\right)\sqrt{7}\right)-\left(-\dfrac{1081}{5}+\left(\dfrac{219}{5}\right)\sqrt{7}\right)\\ &=&-\dfrac{347}{24}+\left(-\dfrac{1859}{60}\right)\sqrt{7}+\dfrac{1081}{5}+\left(-\dfrac{219}{5}\right)\sqrt{7}\\ &=&\dfrac{24209}{120}+\left(-\dfrac{4487}{60}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(-9+\left(-\dfrac{33}{4}\right)\sqrt{175}+\left(\dfrac{7}{3}\right)\sqrt{28}+\left(\dfrac{60}{7}\right)\sqrt{28}+\left(\dfrac{53}{3}\right)\sqrt{28}+\left(-\dfrac{7}{8}\right)\sqrt{49}+\left(\dfrac{65}{6}\right)\sqrt{63}+\left(\left(-\dfrac{34}{5}\right)\sqrt{63}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{22}{7}\right)\sqrt{63}\right)+\left(\left(\dfrac{75}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{175}\right)-\left(\left(5\right)\sqrt{49}\right)-\left(\left(-5\right)\sqrt{49}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{63}\right)\right)\times\left(\dfrac{53}{5}-\left(\left(\dfrac{81}{5}\right)\sqrt{49}+\left(\dfrac{81}{5}\right)\sqrt{49}+\left(-\dfrac{77}{5}\right)\sqrt{28}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{63}\right)-\left(\left(7\right)\sqrt{28}\right)-\left(\left(-\dfrac{19}{5}\right)\sqrt{175}+\left(-\dfrac{19}{3}\right)\sqrt{63}\right)\right)\\ &=&\left(-9+\left(-\dfrac{165}{4}\right)\sqrt{7}+\left(\dfrac{14}{3}\right)\sqrt{7}+\left(\dfrac{120}{7}\right)\sqrt{7}+\left(\dfrac{106}{3}\right)\sqrt{7}-\dfrac{49}{8}+\left(\dfrac{65}{2}\right)\sqrt{7}+\left(\left(-\dfrac{102}{5}\right)\sqrt{7}\right)+\dfrac{2}{3}-\left(\left(-\dfrac{66}{7}\right)\sqrt{7}\right)+\left(\left(\dfrac{150}{7}\right)\sqrt{7}\right)-\left(\left(\dfrac{145}{2}\right)\sqrt{7}\right)-35+35-\left(\left(\dfrac{52}{3}\right)\sqrt{7}\right)\right)\times\left(\dfrac{53}{5}-\left(\dfrac{567}{5}+\dfrac{567}{5}+\left(-\dfrac{154}{5}\right)\sqrt{7}\right)-\left(\left(11\right)\sqrt{7}\right)-\left(\left(14\right)\sqrt{7}\right)-\left(\left(-19\right)\sqrt{7}+\left(-19\right)\sqrt{7}\right)\right)\\ &=&\left(-\dfrac{347}{24}+\left(-\dfrac{1859}{60}\right)\sqrt{7}\right)\left(-\dfrac{1081}{5}+\left(\dfrac{219}{5}\right)\sqrt{7}\right)\\ &=&\dfrac{375107}{120}+\left(\dfrac{3639193}{600}\right)\sqrt{7}+\left(-\dfrac{135707}{100}\right)\sqrt{49}\\ &=&-\dfrac{3824159}{600}+\left(\dfrac{3639193}{600}\right)\sqrt{7}\\ \end{eqnarray*}