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Exercice
Calculer \[\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{6}{3}}{\dfrac{27}{2}}}{9}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{\dfrac{\dfrac{\dfrac{27}{2}}{\dfrac{27}{2}}}{-5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\]
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Exercice
On a \( X=\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{\dfrac{\dfrac{6}{3}}{\dfrac{27}{2}}}{9}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{\dfrac{\dfrac{\dfrac{27}{2}}{\dfrac{27}{2}}}{-5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}=\dfrac{16833893}{362880}\) . Voici le détail :
\begin{eqnarray*}
X &=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{\dfrac{2}{\dfrac{27}{2}}}{9}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{\dfrac{\dfrac{\dfrac{27}{2}}{\dfrac{27}{2}}}{-5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{\dfrac{4}{27}}{9}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{\dfrac{\dfrac{\dfrac{27}{2}}{\dfrac{27}{2}}}{-5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{4}{243}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{\dfrac{\dfrac{\dfrac{27}{2}}{\dfrac{27}{2}}}{-5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{4}{243}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{\dfrac{1}{-5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{4}{243}+5-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{-\dfrac{1}{5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{\dfrac{1219}{243}-6-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{-\dfrac{1}{5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{-\dfrac{239}{243}-\dfrac{97}{2}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{-\dfrac{1}{5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{-\dfrac{24049}{486}}{\dfrac{\dfrac{\dfrac{\dfrac{8}{3}}{-\dfrac{1}{5}}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{-\dfrac{24049}{486}}{\dfrac{\dfrac{-\dfrac{40}{3}}{-5}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{-\dfrac{24049}{486}}{\dfrac{\dfrac{8}{3}}{\dfrac{1}{8}}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{\dfrac{-\dfrac{24049}{486}}{\dfrac{64}{3}}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8-\dfrac{-\dfrac{24049}{10368}}{7}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+6-\dfrac{8}{3}+8+\dfrac{24049}{72576}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+\dfrac{10}{3}+8+\dfrac{24049}{72576}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+\dfrac{34}{3}+\dfrac{24049}{72576}+\dfrac{108}{5}\\&=&\dfrac{105}{8}+\dfrac{846577}{72576}+\dfrac{108}{5}\\&=&\dfrac{1799137}{72576}+\dfrac{108}{5}\\&=&\dfrac{16833893}{362880}\\
\end{eqnarray*}