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Exercice
Calculer \[-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\left(-7+\dfrac{111}{4}+\dfrac{45}{8}-9\right)-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{\dfrac{\dfrac{123}{11}}{4}}{7}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\]
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Exercice
On a \( X=-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\left(-7+\dfrac{111}{4}+\dfrac{45}{8}-9\right)-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{\dfrac{\dfrac{123}{11}}{4}}{7}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}=-\dfrac{806794481}{117043520}\) . Voici le détail :
\begin{eqnarray*}
X &=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\left(\dfrac{83}{4}+\dfrac{45}{8}-9\right)-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{\dfrac{\dfrac{123}{11}}{4}}{7}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\left(\dfrac{211}{8}-9\right)-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{\dfrac{\dfrac{123}{11}}{4}}{7}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\dfrac{139}{8}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{\dfrac{\dfrac{123}{11}}{4}}{7}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\dfrac{139}{8}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{\dfrac{123}{44}}{7}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\dfrac{139}{8}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{\dfrac{123}{308}}{9}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\dfrac{139}{8}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{41}{924}}}{\dfrac{-7+\dfrac{123}{11}-\dfrac{27}{4}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\dfrac{139}{8}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{41}{924}}}{\dfrac{-7+\dfrac{195}{44}+12+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{\left(-\dfrac{27}{4}\right)\times\dfrac{139}{8}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{41}{924}}}{\dfrac{-7+\dfrac{723}{44}+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{\dfrac{\dfrac{-\dfrac{111}{4}}{-\dfrac{45}{8}}}{4}}{\dfrac{41}{924}}}{\dfrac{-7+\dfrac{723}{44}+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{\dfrac{\dfrac{74}{15}}{4}}{\dfrac{41}{924}}}{\dfrac{-7+\dfrac{723}{44}+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{\dfrac{37}{30}}{\dfrac{41}{924}}}{\dfrac{-7+\dfrac{723}{44}+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{5698}{205}}{\dfrac{-7+\dfrac{723}{44}+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{5698}{205}}{\dfrac{\dfrac{415}{44}+9}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{5698}{205}}{\dfrac{\dfrac{811}{44}}{\dfrac{35}{2}-1}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3753}{32}-7+\dfrac{5698}{205}}{\dfrac{\dfrac{811}{44}}{\dfrac{33}{2}}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{3977}{32}+\dfrac{5698}{205}}{\dfrac{\dfrac{811}{44}}{\dfrac{33}{2}}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{632949}{6560}}{\dfrac{\dfrac{811}{44}}{\dfrac{33}{2}}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{\dfrac{-\dfrac{632949}{6560}}{\dfrac{811}{726}}}{12}\\&=&-\dfrac{78}{11}-7-\dfrac{-\dfrac{229760487}{2660080}}{12}\\&=&-\dfrac{78}{11}-7+\dfrac{76586829}{10640320}\\&=&-\dfrac{155}{11}+\dfrac{76586829}{10640320}\\&=&-\dfrac{806794481}{117043520}\\
\end{eqnarray*}