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 \[\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{\dfrac{\dfrac{85}{6}}{\dfrac{85}{6}}}{3}}-\dfrac{45}{4}-\dfrac{\dfrac{\dfrac{45}{4}}{-\dfrac{14}{5}}}{6}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\]

Cliquer ici pour afficher la solution

Exercice

On a $ X=\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{\dfrac{\dfrac{85}{6}}{\dfrac{85}{6}}}{3}}-\dfrac{45}{4}-\dfrac{\dfrac{\dfrac{45}{4}}{-\dfrac{14}{5}}}{6}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}=-\dfrac{81934050}{23119107840}$ . Voici le détail : \begin{eqnarray*} X &=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\dfrac{\dfrac{\dfrac{45}{4}}{-\dfrac{14}{5}}}{6}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\dfrac{\dfrac{\dfrac{45}{4}}{-\dfrac{14}{5}}}{6}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\dfrac{-\dfrac{225}{56}}{6}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\left(\dfrac{35}{12}-\dfrac{14}{11}\right)\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\dfrac{-4}{\dfrac{7}{3}}\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-\dfrac{3}{2}\right)\times6\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-9\right)\times\dfrac{99}{10}\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-\dfrac{891}{10}\right)\times10}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{34}{7}\times\dfrac{34}{7}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{1156}{49}\times\dfrac{99}{10}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{57222}{245}\times\dfrac{99}{10}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{2832489}{1225}\times\left(\dfrac{85}{6}-\dfrac{45}{4}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{2832489}{1225}\times\left(\dfrac{35}{12}-\dfrac{7}{3}\right)\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{2832489}{1225}\times\dfrac{7}{12}\times\left(-4-4\right)-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{\dfrac{137}{7}}{\dfrac{1}{3}}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{2832489}{1225}\times\dfrac{7}{12}\times-8-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{411}{7}-\dfrac{45}{4}-\left(-\dfrac{75}{112}\right)\times\dfrac{217}{132}\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{2832489}{1225}\times\dfrac{7}{12}\times-8-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{411}{7}-\dfrac{45}{4}-\left(-\dfrac{775}{704}\right)\times\left(-\dfrac{12}{7}\right)\times\left(-891\right)}{\dfrac{2832489}{1225}\times\dfrac{7}{12}\times-8-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{411}{7}-\dfrac{45}{4}-\dfrac{2325}{1232}\times\left(-891\right)}{\dfrac{2832489}{1225}\times\dfrac{7}{12}\times-8-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{411}{7}-\dfrac{45}{4}+\dfrac{188325}{112}}{\dfrac{2832489}{1225}\times\dfrac{7}{12}\times-8-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{411}{7}-\dfrac{45}{4}+\dfrac{188325}{112}}{\dfrac{944163}{700}\times-8-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{85}{6}-\dfrac{411}{7}-\dfrac{45}{4}+\dfrac{188325}{112}}{-\dfrac{1888326}{175}-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{1871}{42}-\dfrac{45}{4}+\dfrac{188325}{112}}{-\dfrac{1888326}{175}-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{-\dfrac{4687}{84}+\dfrac{188325}{112}}{-\dfrac{1888326}{175}-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{546227}{336}}{-\dfrac{1888326}{175}-\dfrac{14}{5}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{546227}{336}}{-\dfrac{1888816}{175}}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{\dfrac{-\dfrac{13655675}{90663168}}{3}}{\dfrac{85}{6}}\\&=&\dfrac{-\dfrac{13655675}{271989504}}{\dfrac{85}{6}}\\&=&-\dfrac{81934050}{23119107840}\\ \end{eqnarray*}