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

Calculer \[\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{\left(-\dfrac{61}{5}\right)\times\dfrac{22}{7}\times2}}{\left(-\dfrac{61}{5}\right)\times11\times\dfrac{22}{7}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{-\dfrac{31}{4}}{-\dfrac{61}{5}}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\]

Cliquer ici pour afficher la solution

Exercice

On a $ X=\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{\left(-\dfrac{61}{5}\right)\times\dfrac{22}{7}\times2}}{\left(-\dfrac{61}{5}\right)\times11\times\dfrac{22}{7}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{-\dfrac{31}{4}}{-\dfrac{61}{5}}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}=\dfrac{7474544214886377216}{23959041906800000}$ . Voici le détail : \begin{eqnarray*} X &=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{\left(-\dfrac{1342}{35}\right)\times2}}{\left(-\dfrac{61}{5}\right)\times11\times\dfrac{22}{7}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{-\dfrac{31}{4}}{-\dfrac{61}{5}}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{-\dfrac{2684}{35}}}{\left(-\dfrac{61}{5}\right)\times11\times\dfrac{22}{7}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{-\dfrac{31}{4}}{-\dfrac{61}{5}}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{-\dfrac{2684}{35}}}{\left(-\dfrac{671}{5}\right)\times\dfrac{22}{7}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{-\dfrac{31}{4}}{-\dfrac{61}{5}}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{-\dfrac{2684}{35}}}{-\dfrac{14762}{35}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{-\dfrac{31}{4}}{-\dfrac{61}{5}}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{-\dfrac{2684}{35}}}{-\dfrac{14762}{35}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\dfrac{4}{5}\times2\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{-\dfrac{2684}{35}}}{-\dfrac{14762}{35}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\dfrac{8}{5}\times\left(-\dfrac{61}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{\dfrac{-\dfrac{61}{5}}{\dfrac{22}{7}}}{-\dfrac{2684}{35}}}{-\dfrac{14762}{35}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\left(-\dfrac{488}{25}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{-\dfrac{427}{110}}{-\dfrac{2684}{35}}}{-\dfrac{14762}{35}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\left(-\dfrac{488}{25}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}+\dfrac{\dfrac{49}{968}}{-\dfrac{14762}{35}}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\left(-\dfrac{488}{25}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}-\dfrac{1715}{14289616}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{\left(-\dfrac{85}{7}-\dfrac{44}{9}\right)-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\left(-\dfrac{488}{25}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}-\dfrac{1715}{14289616}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{155}{244}\times\left(-\dfrac{488}{25}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}-\dfrac{1715}{14289616}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\left(-\dfrac{62}{5}\right)\times\left(-\dfrac{9}{10}\right)\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}-\dfrac{1715}{14289616}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{279}{25}\times11-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}-\dfrac{1715}{14289616}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7+7-10}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{4}{5}-\dfrac{1715}{14289616}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(\dfrac{57149889}{71448080}-\dfrac{85}{7}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}-\left(-\dfrac{5673037577}{500136560}+7\right)}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{1073}{63}-\dfrac{1}{4}}{-1+1-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-1+1-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{0-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3069}{25}-7}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{7-\dfrac{4}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{\dfrac{31}{5}-2+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{\dfrac{21}{5}+10}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{\dfrac{71}{5}}}{7-3}}}{2}\\&=&\dfrac{\dfrac{\dfrac{12-\dfrac{1}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{\dfrac{71}{5}}}{4}}}{2}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{47}{4}+\dfrac{2172081657}{500136560}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{\dfrac{71}{5}}}{4}}}{2}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{32194744948}{2000546240}}{11}}{\dfrac{\dfrac{\dfrac{-\dfrac{4355}{252}}{-\dfrac{3244}{25}}}{\dfrac{71}{5}}}{4}}}{2}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{32194744948}{2000546240}}{11}}{\dfrac{\dfrac{\dfrac{108875}{817488}}{\dfrac{71}{5}}}{4}}}{2}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{32194744948}{2000546240}}{11}}{\dfrac{\dfrac{544375}{58041648}}{4}}}{2}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{32194744948}{2000546240}}{11}}{\dfrac{544375}{232166592}}}{2}\\&=&\dfrac{\dfrac{\dfrac{32194744948}{22006008640}}{\dfrac{544375}{232166592}}}{2}\\&=&\dfrac{\dfrac{7474544214886377216}{11979520953400000}}{2}\\&=&\dfrac{7474544214886377216}{23959041906800000}\\ \end{eqnarray*}