L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Calculer \[\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{\dfrac{11}{\dfrac{37}{9}}}{\dfrac{37}{9}}}{-\dfrac{9}{8}-6}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{37}{9}-2+\dfrac{16}{3}\right)\times\dfrac{12+3+\dfrac{16}{3}}{-9}+5}\]
Cliquer ici pour afficher la solution
Exercice
On a \( X=\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{\dfrac{11}{\dfrac{37}{9}}}{\dfrac{37}{9}}}{-\dfrac{9}{8}-6}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{37}{9}-2+\dfrac{16}{3}\right)\times\dfrac{12+3+\dfrac{16}{3}}{-9}+5}=-\dfrac{139968}{54909221}\) . Voici le détail :
\begin{eqnarray*}
X &=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{\dfrac{99}{37}}{\dfrac{37}{9}}}{-\dfrac{9}{8}-6}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{37}{9}-2+\dfrac{16}{3}\right)\times\dfrac{12+3+\dfrac{16}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{891}{1369}}{-\dfrac{9}{8}-6}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{37}{9}-2+\dfrac{16}{3}\right)\times\dfrac{12+3+\dfrac{16}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{891}{1369}}{-\dfrac{57}{8}}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{37}{9}-2+\dfrac{16}{3}\right)\times\dfrac{12+3+\dfrac{16}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{891}{1369}}{-\dfrac{57}{8}}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{19}{9}+\dfrac{16}{3}\right)\times\dfrac{12+3+\dfrac{16}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{891}{1369}}{-\dfrac{57}{8}}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{19}{9}+\dfrac{16}{3}\right)\times\dfrac{15+\dfrac{16}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{\dfrac{\dfrac{891}{1369}}{-\dfrac{57}{8}}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{19}{9}+\dfrac{16}{3}\right)\times\dfrac{\dfrac{61}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{-\dfrac{2376}{26011}}}{\left(-2\right)\times12\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{19}{9}+\dfrac{16}{3}\right)\times\dfrac{\dfrac{61}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{-\dfrac{2376}{26011}}}{\left(-24\right)\times\left(-2\right)}\times11}}{\left(-9\right)\times\left(\dfrac{19}{9}+\dfrac{16}{3}\right)\times\dfrac{\dfrac{61}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{-\dfrac{2376}{26011}}}{48}\times11}}{\left(-9\right)\times\left(\dfrac{19}{9}+\dfrac{16}{3}\right)\times\dfrac{\dfrac{61}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{-\dfrac{2376}{26011}}}{48}\times11}}{\left(-9\right)\times\dfrac{67}{9}\times\dfrac{\dfrac{61}{3}}{-9}+5}\\&=&\dfrac{\dfrac{3}{\dfrac{\dfrac{3}{-\dfrac{2376}{26011}}}{48}\times11}}{\left(-9\right)\times\dfrac{67}{9}\times\left(-\dfrac{61}{27}\right)+5}\\&=&\dfrac{\dfrac{3}{\dfrac{-\dfrac{26011}{792}}{48}\times11}}{\left(-9\right)\times\dfrac{67}{9}\times\left(-\dfrac{61}{27}\right)+5}\\&=&\dfrac{\dfrac{3}{\left(-\dfrac{26011}{38016}\right)\times11}}{\left(-9\right)\times\dfrac{67}{9}\times\left(-\dfrac{61}{27}\right)+5}\\&=&\dfrac{\dfrac{3}{\left(-\dfrac{26011}{38016}\right)\times11}}{\left(-67\right)\times\left(-\dfrac{61}{27}\right)+5}\\&=&\dfrac{\dfrac{3}{\left(-\dfrac{26011}{38016}\right)\times11}}{\dfrac{4087}{27}+5}\\&=&\dfrac{\dfrac{3}{-\dfrac{26011}{3456}}}{\dfrac{4087}{27}+5}\\&=&\dfrac{\dfrac{3}{-\dfrac{26011}{3456}}}{\dfrac{4222}{27}}\\&=&\dfrac{-\dfrac{10368}{26011}}{\dfrac{4222}{27}}\\&=&-\dfrac{139968}{54909221}\\
\end{eqnarray*}