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=5-\left(\left(\dfrac{10}{3}\right)\sqrt{28}\right)+7-\left(\left(\dfrac{9}{5}\right)\sqrt{49}\right)+\left(-\dfrac{71}{5}\right)\sqrt{28}+\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)+\left(\dfrac{17}{9}\right)\sqrt{28}+\dfrac{19}{4}$ et $ Y=\left(\left(\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{79}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{49}\right)+\dfrac{53}{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{49}+\dfrac{11}{3}\right)$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(5-\left(\left(\dfrac{10}{3}\right)\sqrt{28}\right)+7-\left(\left(\dfrac{9}{5}\right)\sqrt{49}\right)+\left(-\dfrac{71}{5}\right)\sqrt{28}+\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)+\left(\dfrac{17}{9}\right)\sqrt{28}+\dfrac{19}{4}\right)+\left(\left(\left(\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{79}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{49}\right)+\dfrac{53}{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{49}+\dfrac{11}{3}\right)\right)\\ &=&\left(5-\left(\left(\dfrac{20}{3}\right)\sqrt{7}\right)+7-\dfrac{63}{5}+\left(-\dfrac{142}{5}\right)\sqrt{7}+\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)+\left(\dfrac{34}{9}\right)\sqrt{7}+\dfrac{19}{4}\right)+\left(\left(\left(\dfrac{195}{2}\right)\sqrt{7}\right)-\left(\dfrac{553}{9}-\left(\left(-\dfrac{78}{7}\right)\sqrt{7}\right)-\dfrac{287}{8}+\dfrac{53}{9}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{7}\right)-\left(\dfrac{371}{8}+\dfrac{11}{3}\right)\right)\\ &=&5-\left(\left(\dfrac{20}{3}\right)\sqrt{7}\right)+7-\dfrac{63}{5}+\left(-\dfrac{142}{5}\right)\sqrt{7}+\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)+\left(\dfrac{34}{9}\right)\sqrt{7}+\dfrac{19}{4}+\left(\left(\dfrac{195}{2}\right)\sqrt{7}\right)-\left(\dfrac{553}{9}-\left(\left(-\dfrac{78}{7}\right)\sqrt{7}\right)-\dfrac{287}{8}+\dfrac{53}{9}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{7}\right)-\left(\dfrac{371}{8}+\dfrac{11}{3}\right)\\ &=&-\dfrac{1547}{20}+\left(\dfrac{149923}{2520}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(5-\left(\left(\dfrac{10}{3}\right)\sqrt{28}\right)+7-\left(\left(\dfrac{9}{5}\right)\sqrt{49}\right)+\left(-\dfrac{71}{5}\right)\sqrt{28}+\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)+\left(\dfrac{17}{9}\right)\sqrt{28}+\dfrac{19}{4}\right)-\left(\left(\left(\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{79}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{49}\right)+\dfrac{53}{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{49}+\dfrac{11}{3}\right)\right)\\ &=&\left(5-\left(\left(\dfrac{20}{3}\right)\sqrt{7}\right)+7-\dfrac{63}{5}+\left(-\dfrac{142}{5}\right)\sqrt{7}+\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)+\left(\dfrac{34}{9}\right)\sqrt{7}+\dfrac{19}{4}\right)-\left(\left(\left(\dfrac{195}{2}\right)\sqrt{7}\right)-\left(\dfrac{553}{9}-\left(\left(-\dfrac{78}{7}\right)\sqrt{7}\right)-\dfrac{287}{8}+\dfrac{53}{9}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{7}\right)-\left(\dfrac{371}{8}+\dfrac{11}{3}\right)\right)\\ &=&\left(\dfrac{83}{20}+\left(-\dfrac{112}{45}\right)\sqrt{7}\right)-\left(\left(\dfrac{3471}{56}\right)\sqrt{7}-\dfrac{163}{2}\right)\\ &=&\dfrac{83}{20}+\left(-\dfrac{112}{45}\right)\sqrt{7}+\left(-\dfrac{3471}{56}\right)\sqrt{7}+\dfrac{163}{2}\\ &=&\dfrac{1713}{20}+\left(-\dfrac{162467}{2520}\right)\sqrt{7}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(5-\left(\left(\dfrac{10}{3}\right)\sqrt{28}\right)+7-\left(\left(\dfrac{9}{5}\right)\sqrt{49}\right)+\left(-\dfrac{71}{5}\right)\sqrt{28}+\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)-\left(\left(-\dfrac{36}{5}\right)\sqrt{28}\right)+\left(\dfrac{17}{9}\right)\sqrt{28}+\dfrac{19}{4}\right)\times\left(\left(\left(\dfrac{39}{2}\right)\sqrt{175}\right)-\left(\left(\left(\dfrac{79}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{28}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{49}\right)+\dfrac{53}{9}\right)-\left(\left(\dfrac{65}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{53}{8}\right)\sqrt{49}+\dfrac{11}{3}\right)\right)\\ &=&\left(5-\left(\left(\dfrac{20}{3}\right)\sqrt{7}\right)+7-\dfrac{63}{5}+\left(-\dfrac{142}{5}\right)\sqrt{7}+\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)-\left(\left(-\dfrac{72}{5}\right)\sqrt{7}\right)+\left(\dfrac{34}{9}\right)\sqrt{7}+\dfrac{19}{4}\right)\times\left(\left(\left(\dfrac{195}{2}\right)\sqrt{7}\right)-\left(\dfrac{553}{9}-\left(\left(-\dfrac{78}{7}\right)\sqrt{7}\right)-\dfrac{287}{8}+\dfrac{53}{9}\right)-\left(\left(\dfrac{195}{8}\right)\sqrt{7}\right)-\left(\dfrac{371}{8}+\dfrac{11}{3}\right)\right)\\ &=&\left(\dfrac{83}{20}+\left(-\dfrac{112}{45}\right)\sqrt{7}\right)\left(\left(\dfrac{3471}{56}\right)\sqrt{7}-\dfrac{163}{2}\right)\\ &=&\left(\dfrac{4637509}{10080}\right)\sqrt{7}-\dfrac{13529}{40}+\left(-\dfrac{2314}{15}\right)\sqrt{49}\\ &=&\left(\dfrac{4637509}{10080}\right)\sqrt{7}-\dfrac{170171}{120}\\ \end{eqnarray*}