L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{8}\right)-3\right)-\left(\left(\left(\dfrac{17}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{4}\right)-\left(\left(9\right)\sqrt{8}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{18}\right)\) et \( Y=\left(\left(-\dfrac{35}{3}\right)\sqrt{50}-\dfrac{62}{9}+\left(\dfrac{11}{9}\right)\sqrt{4}+\left(-\dfrac{11}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{18}\right)\) . 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(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{8}\right)-3\right)-\left(\left(\left(\dfrac{17}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{4}\right)-\left(\left(9\right)\sqrt{8}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{18}\right)\right)+\left(\left(\left(-\dfrac{35}{3}\right)\sqrt{50}-\dfrac{62}{9}+\left(\dfrac{11}{9}\right)\sqrt{4}+\left(-\dfrac{11}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{5}\right)\sqrt{2}\right)-3\right)-\left(\dfrac{17}{4}-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(18\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{2}\right)\right)+\left(\left(\left(-\dfrac{175}{3}\right)\sqrt{2}-\dfrac{62}{9}+\dfrac{22}{9}-\dfrac{11}{3}\right)-\left(\left(\dfrac{189}{5}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{5}\right)\sqrt{2}\right)-3\right)-\left(\dfrac{17}{4}-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(18\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{2}\right)+\left(\left(-\dfrac{175}{3}\right)\sqrt{2}-\dfrac{62}{9}+\dfrac{22}{9}-\dfrac{11}{3}\right)-\left(\left(\dfrac{189}{5}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{557}{4}\right)\sqrt{2}-\dfrac{617}{36}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{8}\right)-3\right)-\left(\left(\left(\dfrac{17}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{4}\right)-\left(\left(9\right)\sqrt{8}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{18}\right)\right)-\left(\left(\left(-\dfrac{35}{3}\right)\sqrt{50}-\dfrac{62}{9}+\left(\dfrac{11}{9}\right)\sqrt{4}+\left(-\dfrac{11}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{5}\right)\sqrt{2}\right)-3\right)-\left(\dfrac{17}{4}-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(18\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{2}\right)\right)-\left(\left(\left(-\dfrac{175}{3}\right)\sqrt{2}-\dfrac{62}{9}+\dfrac{22}{9}-\dfrac{11}{3}\right)-\left(\left(\dfrac{189}{5}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{2587}{60}\right)\sqrt{2}-\dfrac{325}{36}\right)-\left(\left(-\dfrac{1442}{15}\right)\sqrt{2}-\dfrac{73}{9}\right)\\
&=&\left(-\dfrac{2587}{60}\right)\sqrt{2}-\dfrac{325}{36}+\left(\dfrac{1442}{15}\right)\sqrt{2}+\dfrac{73}{9}\\
&=&\left(\dfrac{3181}{60}\right)\sqrt{2}-\dfrac{11}{12}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{2}{5}\right)\sqrt{8}\right)-3\right)-\left(\left(\left(\dfrac{17}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{6}\right)\sqrt{50}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{4}\right)-\left(\left(9\right)\sqrt{8}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{18}\right)\right)\times\left(\left(\left(-\dfrac{35}{3}\right)\sqrt{50}-\dfrac{62}{9}+\left(\dfrac{11}{9}\right)\sqrt{4}+\left(-\dfrac{11}{6}\right)\sqrt{4}\right)-\left(\left(\dfrac{63}{5}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{117}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{4}{5}\right)\sqrt{2}\right)-3\right)-\left(\dfrac{17}{4}-\left(\left(-\dfrac{115}{6}\right)\sqrt{2}\right)+\dfrac{16}{9}-\left(\left(18\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{2}\right)\right)\times\left(\left(\left(-\dfrac{175}{3}\right)\sqrt{2}-\dfrac{62}{9}+\dfrac{22}{9}-\dfrac{11}{3}\right)-\left(\left(\dfrac{189}{5}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{2587}{60}\right)\sqrt{2}-\dfrac{325}{36}\right)\left(\left(-\dfrac{1442}{15}\right)\sqrt{2}-\dfrac{73}{9}\right)\\
&=&\left(\dfrac{1865227}{450}\right)\sqrt{4}+\left(\dfrac{219167}{180}\right)\sqrt{2}+\dfrac{23725}{324}\\
&=&\dfrac{67741297}{8100}+\left(\dfrac{219167}{180}\right)\sqrt{2}\\
\end{eqnarray*}