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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


Calculer \[\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{103}{2}+\dfrac{112}{9}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{\dfrac{-11}{\dfrac{97}{4}}}{-11}\times\left(\dfrac{125}{7}+\dfrac{49}{6}+\dfrac{80}{7}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{103}{2}+\dfrac{112}{9}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{\dfrac{-11}{\dfrac{97}{4}}}{-11}\times\left(\dfrac{125}{7}+\dfrac{49}{6}+\dfrac{80}{7}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}=\dfrac{2436127905960}{2.819794406615E+19}\) . Voici le détail : \begin{eqnarray*} X &=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{\dfrac{-11}{\dfrac{97}{4}}}{-11}\times\left(\dfrac{125}{7}+\dfrac{49}{6}+\dfrac{80}{7}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{-\dfrac{44}{97}}{-11}\times\left(\dfrac{125}{7}+\dfrac{49}{6}+\dfrac{80}{7}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\left(\dfrac{125}{7}+\dfrac{49}{6}+\dfrac{80}{7}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\left(\dfrac{1093}{42}+\dfrac{80}{7}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{\dfrac{49}{6}}{\dfrac{4}{3}}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{\dfrac{49}{8}}{\dfrac{17}{3}}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{\dfrac{147}{136}}{\dfrac{97}{4}}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{147}{3298}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{122}{3}-11}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(\dfrac{49}{6}-\dfrac{703}{18}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{147}{3298}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{31}{5}\right)\times\dfrac{49}{6}\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{147}{3298}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{1519}{30}\right)\times\dfrac{4}{97}\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{147}{3298}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{3038}{1455}\right)\times\dfrac{1573}{42}\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{147}{3298}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{341341}{4365}\right)\times\dfrac{\dfrac{-\dfrac{85}{12}}{\dfrac{147}{3298}}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{341341}{4365}\right)\times\dfrac{-\dfrac{140165}{882}}{\dfrac{4}{3}}\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{341341}{4365}\right)\times\left(-\dfrac{140165}{1176}\right)\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{\dfrac{125}{7}}{\dfrac{112}{9}}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{341341}{4365}\right)\times\left(-\dfrac{140165}{1176}\right)\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{\dfrac{1125}{784}}{\dfrac{89}{3}}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{341341}{4365}\right)\times\left(-\dfrac{140165}{1176}\right)\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{\dfrac{3375}{69776}}{\dfrac{125}{7}}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{278}{9}\right)\times\left(-\dfrac{341341}{4365}\right)\times\left(-\dfrac{140165}{1176}\right)\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{27}{9968}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\dfrac{94892798}{39285}\times\left(-\dfrac{140165}{1176}\right)\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{27}{9968}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6+\left(-\dfrac{13300649031670}{46199160}\right)\times\dfrac{125}{7}-11+\dfrac{\dfrac{122}{3}}{\dfrac{27}{9968}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6-\dfrac{1662581128958750}{323394120}-11+\dfrac{\dfrac{122}{3}}{\dfrac{27}{9968}}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{6-\dfrac{1662581128958750}{323394120}-11+\dfrac{1216096}{81}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{-\dfrac{1662579188594030}{323394120}-11+\dfrac{1216096}{81}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{-\dfrac{1662582745929350}{323394120}+\dfrac{1216096}{81}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{31}{5}}{-\dfrac{134275924124521830}{26194923720}}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{\dfrac{812042635320}{671379620622609150}}{6}}{\dfrac{7}{3}}\\&=&\dfrac{\dfrac{812042635320}{4028277723735654900}}{\dfrac{7}{3}}\\&=&\dfrac{2436127905960}{2.819794406615E+19}\\ \end{eqnarray*}