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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 \[\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{22}{5}+\dfrac{75}{4}}}{2-\dfrac{1}{5}-6-16}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}+2-\dfrac{5}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\]
Cliquer ici pour afficher la solution

Exercice


On a \( X=\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{22}{5}+\dfrac{75}{4}}}{2-\dfrac{1}{5}-6-16}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}+2-\dfrac{5}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2=0\) . Voici le détail : \begin{eqnarray*} X &=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{2-\dfrac{1}{5}-6-16}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}+2-\dfrac{5}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{\dfrac{9}{5}-6-16}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}+2-\dfrac{5}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{21}{5}-16}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}+2-\dfrac{5}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}+2-\dfrac{5}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{1}{2}-\dfrac{75}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{77}{4}-\dfrac{89}{2}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}+\left(-\dfrac{89}{2}\right)\times2+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89+\dfrac{355}{16}\times\dfrac{57}{16}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89+\dfrac{20235}{256}\times\dfrac{8}{13}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89+\dfrac{20235}{416}\times\left(-\dfrac{344}{13}\right)}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{\dfrac{-\dfrac{344}{13}}{\dfrac{463}{20}}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{-\dfrac{6880}{6019}}{-\dfrac{101}{5}}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{\dfrac{34400}{607919}}{\dfrac{1}{5}}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{172000}{607919}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{\dfrac{32}{5}}}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{172000}{607919}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{8}{13}}}{\dfrac{8}{13}-\dfrac{255}{4}-89-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{172000}{607919}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{8}{13}}}{-\dfrac{3283}{52}-89-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{172000}{607919}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{8}{13}}}{-\dfrac{7911}{52}-\dfrac{870105}{676}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times\dfrac{\dfrac{172000}{607919}}{0}+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{8}{13}}}{-\dfrac{243237}{169}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{8}{13}}}{-\dfrac{243237}{169}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{\dfrac{0}{\dfrac{8}{13}}}{-\dfrac{243237}{169}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{\dfrac{0}{-\dfrac{243237}{169}}}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{307}{2}}}{2+19-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{307}{2}}}{21-\dfrac{57}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{307}{2}}}{\dfrac{279}{16}-\dfrac{1}{5}}\right)\times2\\&=&\left(-\dfrac{5}{2}-\dfrac{8}{13}\times0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{307}{2}}}{\dfrac{1379}{80}}\right)\times2\\&=&\left(-\dfrac{5}{2}-0+\dfrac{5}{2}-\dfrac{\dfrac{\dfrac{\dfrac{187}{7}}{0}}{\dfrac{307}{2}}}{\dfrac{1379}{80}}\right)\times2\\&=&\left(-\dfrac{5}{2}-0+\dfrac{5}{2}-\dfrac{\dfrac{0}{\dfrac{307}{2}}}{\dfrac{1379}{80}}\right)\times2\\&=&\left(-\dfrac{5}{2}-0+\dfrac{5}{2}-\dfrac{0}{\dfrac{1379}{80}}\right)\times2\\&=&\left(-\dfrac{5}{2}-0+\dfrac{5}{2}-0\right)\times2\\&=&\left(-\dfrac{5}{2}+\dfrac{5}{2}-0\right)\times2\\&=&\left(0-0\right)\times2\\&=&0\times2\\&=&0\\ \end{eqnarray*}