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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 \[\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{\dfrac{73}{11}+1-\dfrac{45}{4}-\dfrac{45}{4}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{73}{11}+10-\dfrac{19}{5}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{\dfrac{73}{11}+1-\dfrac{45}{4}-\dfrac{45}{4}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{73}{11}+10-\dfrac{19}{5}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}=\dfrac{4804191531197917650}{70098735352248900}\) . Voici le détail : \begin{eqnarray*} X &=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{\dfrac{84}{11}-\dfrac{45}{4}-\dfrac{45}{4}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{73}{11}+10-\dfrac{19}{5}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{-\dfrac{159}{44}-\dfrac{45}{4}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{73}{11}+10-\dfrac{19}{5}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{-\dfrac{327}{22}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{73}{11}+10-\dfrac{19}{5}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{-\dfrac{327}{22}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{183}{11}-\dfrac{19}{5}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{-\dfrac{327}{22}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\left(\dfrac{706}{55}+\dfrac{127}{10}\right)\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{-\dfrac{327}{22}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\dfrac{2809}{110}\times\dfrac{-\dfrac{1}{4}}{-\dfrac{127}{10}}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{-\dfrac{79}{2}}{-\dfrac{327}{22}}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\dfrac{2809}{110}\times\dfrac{5}{254}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{\dfrac{869}{327}}{\dfrac{93}{7}}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\dfrac{2809}{110}\times\dfrac{5}{254}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{6083}{30411}\right)\times\left(2\times\left(-\dfrac{1}{4}\right)+\dfrac{45}{4}-\dfrac{2809}{110}\times\dfrac{5}{254}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{6083}{30411}\right)\times\left(-\dfrac{1}{2}+\dfrac{45}{4}-\dfrac{2809}{110}\times\dfrac{5}{254}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{\left(-\dfrac{45}{4}\right)\times\left(-4\right)}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{6083}{30411}\right)\times\left(-\dfrac{1}{2}+\dfrac{45}{4}-\dfrac{2809}{5588}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{45}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\left(\dfrac{29}{5}+\dfrac{6083}{30411}\right)\times\left(-\dfrac{1}{2}+\dfrac{45}{4}-\dfrac{2809}{5588}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{45}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\left(-\dfrac{1}{2}+\dfrac{45}{4}-\dfrac{2809}{5588}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{45}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\left(\dfrac{43}{4}-\dfrac{2809}{5588}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{45}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\left(\dfrac{28631}{2794}+\dfrac{1}{4}\right)+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{45}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\dfrac{58659}{5588}+\dfrac{73}{11}+\dfrac{\dfrac{\left(-4\right)\times1}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{\dfrac{45}{-\dfrac{127}{10}}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\dfrac{58659}{5588}+\dfrac{73}{11}+\dfrac{\dfrac{-4}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{-\dfrac{450}{127}}{-\dfrac{1}{4}}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\dfrac{58659}{5588}+\dfrac{73}{11}+\dfrac{\dfrac{-4}{\dfrac{93}{7}}}{-10}\\&=&\dfrac{\dfrac{1800}{127}}{-\dfrac{127}{10}}+\dfrac{912334}{152055}\times\dfrac{58659}{5588}+\dfrac{73}{11}+\dfrac{\dfrac{-4}{\dfrac{93}{7}}}{-10}\\&=&-\dfrac{18000}{16129}+\dfrac{912334}{152055}\times\dfrac{58659}{5588}+\dfrac{73}{11}+\dfrac{\dfrac{-4}{\dfrac{93}{7}}}{-10}\\&=&-\dfrac{18000}{16129}+\dfrac{53516600106}{849683340}+\dfrac{73}{11}+\dfrac{\dfrac{-4}{\dfrac{93}{7}}}{-10}\\&=&-\dfrac{18000}{16129}+\dfrac{53516600106}{849683340}+\dfrac{73}{11}+\dfrac{-\dfrac{28}{93}}{-10}\\&=&-\dfrac{18000}{16129}+\dfrac{53516600106}{849683340}+\dfrac{73}{11}+\dfrac{14}{465}\\&=&\dfrac{847874942989674}{13704542590860}+\dfrac{73}{11}+\dfrac{14}{465}\\&=&\dfrac{10327055982019194}{150749968499460}+\dfrac{14}{465}\\&=&\dfrac{4804191531197917650}{70098735352248900}\\ \end{eqnarray*}