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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 \[\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\left(-\dfrac{34}{3}+\dfrac{34}{3}-5-6\right)-0+\dfrac{134}{9}-\left(-\dfrac{131}{7}-6+\dfrac{34}{3}\right)+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\left(-\dfrac{34}{3}+\dfrac{34}{3}-5-6\right)-0+\dfrac{134}{9}-\left(-\dfrac{131}{7}-6+\dfrac{34}{3}\right)+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}=\dfrac{128137500}{262109408903066736}\) . Voici le détail : \begin{eqnarray*} X &=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\left(0-5-6\right)-0+\dfrac{134}{9}-\left(-\dfrac{131}{7}-6+\dfrac{34}{3}\right)+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\left(-5-6\right)-0+\dfrac{134}{9}-\left(-\dfrac{131}{7}-6+\dfrac{34}{3}\right)+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}-\left(-\dfrac{131}{7}-6+\dfrac{34}{3}\right)+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}-\left(-\dfrac{173}{7}+\dfrac{34}{3}\right)+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{81}{8}\times5\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{\dfrac{405}{8}\times\left(-\dfrac{34}{3}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{\left(-\dfrac{2295}{4}\right)\times0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-6\times9\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-54\times\dfrac{81}{8}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{6}{5}\times\left(-\dfrac{34}{3}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\left(-\dfrac{68}{5}\right)\times\left(-\dfrac{134}{9}\right)}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11-0+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}-11+\dfrac{134}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{35}{9}+\dfrac{281}{21}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+\dfrac{\dfrac{\dfrac{6}{9}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+\dfrac{\dfrac{\dfrac{2}{3}}{0}}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+\dfrac{0}{6}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6+\dfrac{6}{5}-\dfrac{93}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{81}{10}+\dfrac{40}{9}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{329}{90}-\dfrac{2187}{4}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}+\dfrac{-\dfrac{40}{9}}{\dfrac{9112}{45}}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+\dfrac{9}{1}+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0\times\dfrac{132}{5}}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times9\times9\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times81\times\left(-\dfrac{34}{3}\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\left(\dfrac{23}{7}+\dfrac{1088}{63}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\left(\dfrac{185}{9}+0+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\left(\dfrac{185}{9}+\dfrac{93}{10}\right)}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\dfrac{2687}{90}}}{\dfrac{132}{5}}}{6-\dfrac{99073}{180}-\dfrac{25}{1139}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\dfrac{2687}{90}}}{\dfrac{132}{5}}}{6-\dfrac{112848647}{205020}+9+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\dfrac{2687}{90}}}{\dfrac{132}{5}}}{6-\dfrac{111003467}{205020}+\dfrac{6}{0}}\\&=&\dfrac{\dfrac{\dfrac{5}{\dfrac{132}{5}\times\left(-918\right)\times\dfrac{2687}{90}}}{\dfrac{132}{5}}}{6-\dfrac{111003467}{205020}+0}\\&=&\dfrac{\dfrac{\dfrac{5}{\left(-\dfrac{121176}{5}\right)\times\dfrac{2687}{90}}}{\dfrac{132}{5}}}{6-\dfrac{111003467}{205020}+0}\\&=&\dfrac{\dfrac{\dfrac{5}{-\dfrac{18088884}{25}}}{\dfrac{132}{5}}}{6-\dfrac{111003467}{205020}+0}\\&=&\dfrac{\dfrac{\dfrac{5}{-\dfrac{18088884}{25}}}{\dfrac{132}{5}}}{-\dfrac{109773347}{205020}+0}\\&=&\dfrac{\dfrac{\dfrac{5}{-\dfrac{18088884}{25}}}{\dfrac{132}{5}}}{-\dfrac{109773347}{205020}}\\&=&\dfrac{\dfrac{-\dfrac{125}{18088884}}{\dfrac{132}{5}}}{-\dfrac{109773347}{205020}}\\&=&\dfrac{-\dfrac{625}{2387732688}}{-\dfrac{109773347}{205020}}\\&=&\dfrac{128137500}{262109408903066736}\\ \end{eqnarray*}