L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{21}{5}\right)\sqrt{8}-\dfrac{47}{9}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{37}{2}\right)\sqrt{4}+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{4}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{62}{9}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{18}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{23}{5}\right)\sqrt{8}+\left(-\dfrac{63}{4}\right)\sqrt{4}\right)\) et \( Y=\left(\dfrac{1}{3}\right)\sqrt{50}+\dfrac{49}{3}+\left(\dfrac{5}{2}\right)\sqrt{8}+\left(-\dfrac{38}{3}\right)\sqrt{4}+\left(\left(-\dfrac{19}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)+\left(-\dfrac{17}{2}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}-\dfrac{47}{9}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{37}{2}\right)\sqrt{4}+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{4}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{62}{9}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{18}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{23}{5}\right)\sqrt{8}+\left(-\dfrac{63}{4}\right)\sqrt{4}\right)\right)+\left(\left(\dfrac{1}{3}\right)\sqrt{50}+\dfrac{49}{3}+\left(\dfrac{5}{2}\right)\sqrt{8}+\left(-\dfrac{38}{3}\right)\sqrt{4}+\left(\left(-\dfrac{19}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)+\left(-\dfrac{17}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{57}{2}+37+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{2}\right)-\left(\dfrac{20}{3}+\dfrac{57}{2}+\dfrac{124}{9}+\left(-\dfrac{63}{5}\right)\sqrt{2}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{46}{5}\right)\sqrt{2}-\dfrac{63}{2}\right)\right)+\left(\left(\dfrac{5}{3}\right)\sqrt{2}+\dfrac{49}{3}+\left(5\right)\sqrt{2}-\dfrac{76}{3}+\left(\left(-19\right)\sqrt{2}\right)-27-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)+\left(-\dfrac{85}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{42}{5}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{57}{2}+37+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{2}\right)-\left(\dfrac{20}{3}+\dfrac{57}{2}+\dfrac{124}{9}+\left(-\dfrac{63}{5}\right)\sqrt{2}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{46}{5}\right)\sqrt{2}-\dfrac{63}{2}\right)+\left(\dfrac{5}{3}\right)\sqrt{2}+\dfrac{49}{3}+\left(5\right)\sqrt{2}-\dfrac{76}{3}+\left(\left(-19\right)\sqrt{2}\right)-27-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)+\left(-\dfrac{85}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{1207}{12}\right)\sqrt{2}+\dfrac{301}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}-\dfrac{47}{9}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{37}{2}\right)\sqrt{4}+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{4}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{62}{9}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{18}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{23}{5}\right)\sqrt{8}+\left(-\dfrac{63}{4}\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{1}{3}\right)\sqrt{50}+\dfrac{49}{3}+\left(\dfrac{5}{2}\right)\sqrt{8}+\left(-\dfrac{38}{3}\right)\sqrt{4}+\left(\left(-\dfrac{19}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)+\left(-\dfrac{17}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{57}{2}+37+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{2}\right)-\left(\dfrac{20}{3}+\dfrac{57}{2}+\dfrac{124}{9}+\left(-\dfrac{63}{5}\right)\sqrt{2}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{46}{5}\right)\sqrt{2}-\dfrac{63}{2}\right)\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{2}+\dfrac{49}{3}+\left(5\right)\sqrt{2}-\dfrac{76}{3}+\left(\left(-19\right)\sqrt{2}\right)-27-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)+\left(-\dfrac{85}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{29}{2}\right)\sqrt{2}+\dfrac{949}{18}\right)-\left(\left(-\dfrac{1033}{12}\right)\sqrt{2}-36\right)\\
&=&\left(-\dfrac{29}{2}\right)\sqrt{2}+\dfrac{949}{18}+\left(\dfrac{1033}{12}\right)\sqrt{2}+36\\
&=&\left(\dfrac{859}{12}\right)\sqrt{2}+\dfrac{1597}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{21}{5}\right)\sqrt{8}-\dfrac{47}{9}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{37}{2}\right)\sqrt{4}+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{10}{3}\right)\sqrt{4}+\left(\dfrac{57}{4}\right)\sqrt{4}+\left(\dfrac{62}{9}\right)\sqrt{4}+\left(-\dfrac{21}{5}\right)\sqrt{18}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{23}{5}\right)\sqrt{8}+\left(-\dfrac{63}{4}\right)\sqrt{4}\right)\right)\times\left(\left(\dfrac{1}{3}\right)\sqrt{50}+\dfrac{49}{3}+\left(\dfrac{5}{2}\right)\sqrt{8}+\left(-\dfrac{38}{3}\right)\sqrt{4}+\left(\left(-\dfrac{19}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)+\left(-\dfrac{17}{2}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-\dfrac{42}{5}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{57}{2}+37+\dfrac{14}{3}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{2}\right)-\left(\dfrac{20}{3}+\dfrac{57}{2}+\dfrac{124}{9}+\left(-\dfrac{63}{5}\right)\sqrt{2}\right)-\left(-\dfrac{47}{9}+\left(\dfrac{46}{5}\right)\sqrt{2}-\dfrac{63}{2}\right)\right)\times\left(\left(\dfrac{5}{3}\right)\sqrt{2}+\dfrac{49}{3}+\left(5\right)\sqrt{2}-\dfrac{76}{3}+\left(\left(-19\right)\sqrt{2}\right)-27-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)+\left(-\dfrac{85}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{29}{2}\right)\sqrt{2}+\dfrac{949}{18}\right)\left(\left(-\dfrac{1033}{12}\right)\sqrt{2}-36\right)\\
&=&\left(\dfrac{29957}{24}\right)\sqrt{4}+\left(-\dfrac{867565}{216}\right)\sqrt{2}-1898\\
&=&\dfrac{7181}{12}+\left(-\dfrac{867565}{216}\right)\sqrt{2}\\
\end{eqnarray*}