L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Calculer \[\dfrac{\dfrac{\left(-\dfrac{20}{3}+6-4+7-\dfrac{42}{5}+\dfrac{20}{3}+\dfrac{75}{4}+\left(-7\right)\times11\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\]
Cliquer ici pour afficher la solution
Exercice
On a \( X=\dfrac{\dfrac{\left(-\dfrac{20}{3}+6-4+7-\dfrac{42}{5}+\dfrac{20}{3}+\dfrac{75}{4}+\left(-7\right)\times11\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}=-\dfrac{13067}{3780}\) . Voici le détail :
\begin{eqnarray*}
X &=&\dfrac{\dfrac{\left(-\dfrac{2}{3}-4+7-\dfrac{42}{5}+\dfrac{20}{3}+\dfrac{75}{4}+\left(-7\right)\times11\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+3-\dfrac{42}{5}+\dfrac{20}{3}+\dfrac{75}{4}+\left(-7\right)\times11\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}-\dfrac{27}{5}+\dfrac{20}{3}+\dfrac{75}{4}+\left(-7\right)\times11\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+\left(-7\right)\times11\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+\left(-77\right)\times11\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+\left(-847\right)\times\left(-1\right)+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9+\left(-\dfrac{20}{3}\right)\times10\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9+\left(-\dfrac{200}{3}\right)\times\left(-4\right)\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9+\dfrac{800}{3}\times\left(-4\right)+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9-\dfrac{3200}{3}+\left(-\dfrac{20}{3}-\dfrac{42}{5}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9-\dfrac{3200}{3}+\left(-\dfrac{226}{15}-\dfrac{42}{5}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9-\dfrac{3200}{3}+\left(-\dfrac{352}{15}+8\right)+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{2}{3}+\dfrac{19}{15}+\dfrac{75}{4}+847+10-1-12-6-9-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{3}{5}+\dfrac{75}{4}+847+10-1-12-6-9-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+847+10-1-12-6-9-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+857-1-12-6-9-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+856-12-6-9-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+856-18-9-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+856-27-\dfrac{3200}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+856-\dfrac{3281}{3}-\dfrac{232}{15}+7\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{387}{20}+856-\dfrac{3281}{3}-\dfrac{127}{15}\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(\dfrac{17507}{20}-\dfrac{3281}{3}-\dfrac{127}{15}\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{\left(-\dfrac{13099}{60}-\dfrac{127}{15}\right)+9}{-7}}{-9}\\&=&\dfrac{\dfrac{-\dfrac{13607}{60}+9}{-7}}{-9}\\&=&\dfrac{\dfrac{-\dfrac{13067}{60}}{-7}}{-9}\\&=&\dfrac{\dfrac{13067}{420}}{-9}\\&=&-\dfrac{13067}{3780}\\
\end{eqnarray*}