L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Calculer \[\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{92}{3}-\dfrac{65}{4}-\dfrac{92}{3}\right)\times\left(\dfrac{40}{3}+\dfrac{59}{2}\right)\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\left(-10-2+\dfrac{40}{3}\right)\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\]
Cliquer ici pour afficher la solution
Exercice
On a \( X=\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{92}{3}-\dfrac{65}{4}-\dfrac{92}{3}\right)\times\left(\dfrac{40}{3}+\dfrac{59}{2}\right)\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\left(-10-2+\dfrac{40}{3}\right)\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}=\dfrac{5063984}{6825}\) . Voici le détail :
\begin{eqnarray*}
X &=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{563}{12}-\dfrac{92}{3}\right)\times\left(\dfrac{40}{3}+\dfrac{59}{2}\right)\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\left(-10-2+\dfrac{40}{3}\right)\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\left(\dfrac{40}{3}+\dfrac{59}{2}\right)\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\left(-10-2+\dfrac{40}{3}\right)\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\left(-10-2+\dfrac{40}{3}\right)\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\left(-12+\dfrac{40}{3}\right)\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\left(\dfrac{4}{5}+\dfrac{4}{5}\right)}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\dfrac{8}{5}}{\dfrac{-\dfrac{106}{11}+2}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{\dfrac{33}{5}}{11}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\dfrac{8}{5}}{\dfrac{-\dfrac{84}{11}}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\dfrac{\dfrac{3}{5}}{-\dfrac{65}{4}}\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\dfrac{8}{5}}{\dfrac{-\dfrac{84}{11}}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\left(-\dfrac{12}{325}\right)\times\left(-\dfrac{931}{12}\right)\times\dfrac{257}{6}\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\dfrac{8}{5}}{\dfrac{-\dfrac{84}{11}}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\left(-\dfrac{12}{325}\right)\times\left(-\dfrac{239267}{72}\right)\times6\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\dfrac{8}{5}}{\dfrac{-\dfrac{84}{11}}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\left(-\dfrac{12}{325}\right)\times\left(-\dfrac{239267}{12}\right)\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{4}{3}\times\dfrac{8}{5}}{\dfrac{-\dfrac{84}{11}}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\left(-\dfrac{12}{325}\right)\times\left(-\dfrac{239267}{12}\right)\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{32}{15}}{\dfrac{-\dfrac{84}{11}}{\dfrac{33}{5}}}}{-8}\\&=&\dfrac{\left(-\dfrac{12}{325}\right)\times\left(-\dfrac{239267}{12}\right)\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{32}{15}}{-\dfrac{140}{121}}}{-8}\\&=&\dfrac{\dfrac{239267}{325}\times\left(-\dfrac{65}{4}\right)}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{32}{15}}{-\dfrac{140}{121}}}{-8}\\&=&\dfrac{-\dfrac{239267}{20}}{-\dfrac{65}{4}}+6-\dfrac{\dfrac{\dfrac{32}{15}}{-\dfrac{140}{121}}}{-8}\\&=&\dfrac{-\dfrac{239267}{20}}{-\dfrac{65}{4}}+6-\dfrac{-\dfrac{968}{525}}{-8}\\&=&\dfrac{-\dfrac{239267}{20}}{-\dfrac{65}{4}}+6-\dfrac{121}{525}\\&=&\dfrac{239267}{325}+6-\dfrac{121}{525}\\&=&\dfrac{239267}{325}+\dfrac{3029}{525}\\&=&\dfrac{5063984}{6825}\\
\end{eqnarray*}