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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.

Exercice


Calculer \[\dfrac{7}{2}+\left(-\dfrac{27}{4}+12+\dfrac{94}{11}-12-1+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times\left(-8\right)\times8\times\left(-\dfrac{27}{4}\right)\times12\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\dfrac{7}{2}+\left(-\dfrac{27}{4}+12+\dfrac{94}{11}-12-1+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times\left(-8\right)\times8\times\left(-\dfrac{27}{4}\right)\times12\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)=\dfrac{536845805122}{79860}\) . Voici le détail : \begin{eqnarray*} X &=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{38}{11}-1+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times\left(-8\right)\times8\times\left(-\dfrac{27}{4}\right)\times12\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times\left(-8\right)\times8\times\left(-\dfrac{27}{4}\right)\times12\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times\left(-64\right)\times\left(-\dfrac{27}{4}\right)\times12\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times432\times12\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times5184\times\left(\dfrac{94}{11}+10+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times5184\times\left(\dfrac{204}{11}+\dfrac{27}{4}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times5184\times\left(\dfrac{1113}{44}+8\right)-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+12-\dfrac{49}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times5184\times\dfrac{1465}{44}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+\dfrac{83}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+8-1\right)-12+\left(\left(-1\right)\times5184\times\dfrac{1465}{44}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+\dfrac{83}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+7\right)-12+\left(\left(-1\right)\times5184\times\dfrac{1465}{44}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+\dfrac{83}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+7\right)-12+\left(\left(-5184\right)\times\dfrac{1465}{44}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(-\dfrac{27}{4}+\dfrac{83}{11}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+7\right)-12+\left(-\dfrac{1898640}{11}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(\dfrac{35}{44}+\dfrac{78}{5}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+7\right)-12+\left(-\dfrac{1898640}{11}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\left(\dfrac{3607}{220}+\dfrac{7}{3}\right)+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+7\right)-12+\left(-\dfrac{1898640}{11}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\dfrac{12361}{660}+1-\left(-\dfrac{27}{4}+\dfrac{78}{5}+7\right)-12+\left(-\dfrac{1898640}{11}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\dfrac{12361}{660}+1-\left(\dfrac{177}{20}+7\right)-12+\left(-\dfrac{1898640}{11}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\dfrac{12361}{660}+1-\dfrac{317}{20}-12+\left(-\dfrac{1898640}{11}-\dfrac{11}{2}\right)\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\dfrac{12361}{660}+1-\dfrac{317}{20}-12-\dfrac{3797401}{22}\times\left(-12\right)\times\left(\dfrac{94}{11}-\dfrac{53}{10}\right)\\&=&\dfrac{7}{2}+\dfrac{12361}{660}+1-\dfrac{317}{20}-12-\dfrac{3797401}{22}\times\left(-12\right)\times\dfrac{357}{110}\\&=&\dfrac{7}{2}+\dfrac{13021}{660}-\dfrac{317}{20}-12-\dfrac{3797401}{22}\times\left(-12\right)\times\dfrac{357}{110}\\&=&\dfrac{7}{2}+\dfrac{128}{33}-12-\dfrac{3797401}{22}\times\left(-12\right)\times\dfrac{357}{110}\\&=&\dfrac{7}{2}-\dfrac{268}{33}-\dfrac{3797401}{22}\times\left(-12\right)\times\dfrac{357}{110}\\&=&\dfrac{7}{2}-\dfrac{268}{33}+\dfrac{22784406}{11}\times\dfrac{357}{110}\\&=&\dfrac{7}{2}-\dfrac{268}{33}+\dfrac{8134032942}{1210}\\&=&-\dfrac{305}{66}+\dfrac{8134032942}{1210}\\&=&\dfrac{536845805122}{79860}\\ \end{eqnarray*}