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Exercice
Calculer \[\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(\left(-5\right)\times12\times5-7-\left(-5-5-7\right)\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}+\dfrac{6}{-\dfrac{28}{3}}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\]
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Exercice
On a \( X=\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(\left(-5\right)\times12\times5-7-\left(-5-5-7\right)\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}+\dfrac{6}{-\dfrac{28}{3}}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}=\dfrac{104180592180}{921323768534280}\) . Voici le détail :
\begin{eqnarray*}
X &=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(\left(-60\right)\times5-7-\left(-5-5-7\right)\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}+\dfrac{6}{-\dfrac{28}{3}}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(-300-7-\left(-5-5-7\right)\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}+\dfrac{6}{-\dfrac{28}{3}}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(-300-7-\left(-10-7\right)\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}+\dfrac{6}{-\dfrac{28}{3}}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(-300-7+17\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}+\dfrac{6}{-\dfrac{28}{3}}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(-300-7+17\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-\left(-307+17\right)\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7+290\times\left(-\dfrac{82}{5}+10-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7+290\times\left(-\dfrac{32}{5}-\dfrac{103}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7+290\times\left(-\dfrac{167}{10}\right)-\dfrac{\dfrac{\dfrac{-\dfrac{82}{5}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7+290\times\left(-\dfrac{167}{10}\right)-\dfrac{\dfrac{\dfrac{164}{103}}{-\dfrac{103}{10}}}{-\dfrac{103}{10}+\dfrac{107}{8}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7+290\times\left(-\dfrac{167}{10}\right)-\dfrac{\dfrac{\dfrac{164}{103}}{-\dfrac{103}{10}}}{\dfrac{123}{40}-\dfrac{9}{14}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7+290\times\left(-\dfrac{167}{10}\right)-\dfrac{\dfrac{\dfrac{164}{103}}{-\dfrac{103}{10}}}{\dfrac{681}{280}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-4843-\dfrac{\dfrac{\dfrac{164}{103}}{-\dfrac{103}{10}}}{\dfrac{681}{280}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-4843-\dfrac{-\dfrac{1640}{10609}}{\dfrac{681}{280}}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-4843+\dfrac{459200}{7224729}}}{\dfrac{-5}{-\dfrac{28}{3}}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{7-4843+\dfrac{459200}{7224729}}}{\dfrac{15}{28}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{-4836+\dfrac{459200}{7224729}}}{\dfrac{15}{28}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{-\dfrac{34938330244}{7224729}}}{\dfrac{15}{28}+5+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{-\dfrac{34938330244}{7224729}}}{\dfrac{155}{28}+\dfrac{133}{10}}\\&=&\dfrac{\dfrac{-\dfrac{103}{10}}{-\dfrac{34938330244}{7224729}}}{\dfrac{2637}{140}}\\&=&\dfrac{\dfrac{744147087}{349383302440}}{\dfrac{2637}{140}}\\&=&\dfrac{104180592180}{921323768534280}\\
\end{eqnarray*}