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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 \[\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\left(-2\right)\times\dfrac{27}{11}+\dfrac{49}{4}+2+\dfrac{\dfrac{\dfrac{9}{\dfrac{49}{4}}}{-\dfrac{23}{6}}}{\dfrac{27}{11}}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\left(-2\right)\times\dfrac{27}{11}+\dfrac{49}{4}+2+\dfrac{\dfrac{\dfrac{9}{\dfrac{49}{4}}}{-\dfrac{23}{6}}}{\dfrac{27}{11}}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)=\dfrac{82714531889}{570286794}\) . Voici le détail : \begin{eqnarray*} X &=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{49}{4}+2+\dfrac{\dfrac{\dfrac{9}{\dfrac{49}{4}}}{-\dfrac{23}{6}}}{\dfrac{27}{11}}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}+\dfrac{\dfrac{\dfrac{9}{\dfrac{49}{4}}}{-\dfrac{23}{6}}}{\dfrac{27}{11}}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}+\dfrac{\dfrac{\dfrac{36}{49}}{-\dfrac{23}{6}}}{\dfrac{27}{11}}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}+\dfrac{-\dfrac{216}{1127}}{\dfrac{27}{11}}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{\dfrac{11}{2}}{\dfrac{17}{2}}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{\dfrac{11}{17}}{\dfrac{49}{4}}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{\dfrac{44}{833}}{-\dfrac{41}{12}}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}-\left(-6+2+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}-\left(-4+\dfrac{11}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}-\left(\dfrac{3}{2}-\dfrac{68}{5}\right)+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{23}{6}+7+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9+\dfrac{19}{6}+2-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9+\dfrac{31}{6}-\dfrac{11}{2}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(-\dfrac{54}{11}+\dfrac{57}{4}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{411}{44}-\dfrac{88}{1127}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{459325}{49588}+\dfrac{11}{2}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{732059}{49588}+3\times3-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{732059}{49588}+9-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}-\dfrac{528}{34153}+\dfrac{121}{10}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{732059}{49588}+9-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{4127233}{341530}+7-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{732059}{49588}+9-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{6517943}{341530}-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(9-\dfrac{1}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{732059}{49588}+9-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{6517943}{341530}-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(\dfrac{26}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{1178351}{49588}-6-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{6517943}{341530}-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(\dfrac{26}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\left(-\dfrac{23}{6}+\dfrac{27}{11}+\dfrac{6517943}{341530}-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(\dfrac{26}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\left(-\dfrac{91}{66}+\dfrac{6517943}{341530}-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(\dfrac{26}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\left(\dfrac{99776252}{5635245}-\dfrac{68}{5}\right)\times\left(-7\right)\times\left(\dfrac{26}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\dfrac{4627384}{1127049}\times\left(-7\right)\times\left(\dfrac{26}{3}-\dfrac{41}{12}\right)\\&=&\left(\dfrac{102}{11}+7-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\dfrac{4627384}{1127049}\times\left(-7\right)\times\dfrac{21}{4}\\&=&\left(\dfrac{179}{11}-\dfrac{11}{2}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\dfrac{4627384}{1127049}\times\left(-7\right)\times\dfrac{21}{4}\\&=&\left(\dfrac{237}{22}+\dfrac{41}{12}\right)-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\dfrac{4627384}{1127049}\times\left(-7\right)\times\dfrac{21}{4}\\&=&\dfrac{1873}{132}-\left(\dfrac{880823}{49588}-7\right)-\dfrac{102}{11}-\dfrac{4627384}{1127049}\times\left(-7\right)\times\dfrac{21}{4}\\&=&\dfrac{1873}{132}-\dfrac{533707}{49588}-\dfrac{102}{11}-\dfrac{4627384}{1127049}\times\left(-7\right)\times\dfrac{21}{4}\\&=&\dfrac{1873}{132}-\dfrac{533707}{49588}-\dfrac{102}{11}-\left(-\dfrac{4627384}{161007}\right)\times\dfrac{21}{4}\\&=&\dfrac{1873}{132}-\dfrac{533707}{49588}-\dfrac{102}{11}+\dfrac{1156846}{7667}\\&=&\dfrac{254875}{74382}-\dfrac{102}{11}+\dfrac{1156846}{7667}\\&=&-\dfrac{434849}{74382}+\dfrac{1156846}{7667}\\&=&\dfrac{82714531889}{570286794}\\ \end{eqnarray*}