L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-8\right)\sqrt{8}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{26}{3}\right)\sqrt{50}\right)-\left(\left(2\right)\sqrt{4}\right)-\dfrac{27}{7}-3+\left(\dfrac{73}{7}\right)\sqrt{50}+\left(\left(\dfrac{17}{3}\right)\sqrt{8}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{49}{6}\right)\sqrt{50}+\left(\dfrac{19}{2}\right)\sqrt{50}+\left(\dfrac{37}{2}\right)\sqrt{4}\) et \( Y=\dfrac{69}{5}\) . 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(-8\right)\sqrt{8}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{26}{3}\right)\sqrt{50}\right)-\left(\left(2\right)\sqrt{4}\right)-\dfrac{27}{7}-3+\left(\dfrac{73}{7}\right)\sqrt{50}+\left(\left(\dfrac{17}{3}\right)\sqrt{8}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{49}{6}\right)\sqrt{50}+\left(\dfrac{19}{2}\right)\sqrt{50}+\left(\dfrac{37}{2}\right)\sqrt{4}\right)+\left(\dfrac{69}{5}\right)\\
&=&\left(\left(\left(-16\right)\sqrt{2}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{130}{3}\right)\sqrt{2}\right)-4-\dfrac{27}{7}-3+\left(\dfrac{365}{7}\right)\sqrt{2}+\left(\left(\dfrac{34}{3}\right)\sqrt{2}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{245}{6}\right)\sqrt{2}+\left(\dfrac{95}{2}\right)\sqrt{2}+37\right)+\left(\dfrac{69}{5}\right)\\
&=&\left(\left(-16\right)\sqrt{2}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{130}{3}\right)\sqrt{2}\right)-4-\dfrac{27}{7}-3+\left(\dfrac{365}{7}\right)\sqrt{2}+\left(\left(\dfrac{34}{3}\right)\sqrt{2}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{245}{6}\right)\sqrt{2}+\left(\dfrac{95}{2}\right)\sqrt{2}+37+\dfrac{69}{5}\\
&=&\left(\dfrac{2047}{21}\right)\sqrt{2}+\dfrac{16657}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-8\right)\sqrt{8}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{26}{3}\right)\sqrt{50}\right)-\left(\left(2\right)\sqrt{4}\right)-\dfrac{27}{7}-3+\left(\dfrac{73}{7}\right)\sqrt{50}+\left(\left(\dfrac{17}{3}\right)\sqrt{8}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{49}{6}\right)\sqrt{50}+\left(\dfrac{19}{2}\right)\sqrt{50}+\left(\dfrac{37}{2}\right)\sqrt{4}\right)-\left(\dfrac{69}{5}\right)\\
&=&\left(\left(\left(-16\right)\sqrt{2}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{130}{3}\right)\sqrt{2}\right)-4-\dfrac{27}{7}-3+\left(\dfrac{365}{7}\right)\sqrt{2}+\left(\left(\dfrac{34}{3}\right)\sqrt{2}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{245}{6}\right)\sqrt{2}+\left(\dfrac{95}{2}\right)\sqrt{2}+37\right)-\left(\dfrac{69}{5}\right)\\
&=&\left(\left(\dfrac{2047}{21}\right)\sqrt{2}+\dfrac{2462}{63}\right)-\left(\dfrac{69}{5}\right)\\
&=&\left(\dfrac{2047}{21}\right)\sqrt{2}+\dfrac{2462}{63}+-\dfrac{69}{5}\\
&=&\left(\dfrac{2047}{21}\right)\sqrt{2}+\dfrac{7963}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-8\right)\sqrt{8}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{26}{3}\right)\sqrt{50}\right)-\left(\left(2\right)\sqrt{4}\right)-\dfrac{27}{7}-3+\left(\dfrac{73}{7}\right)\sqrt{50}+\left(\left(\dfrac{17}{3}\right)\sqrt{8}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{49}{6}\right)\sqrt{50}+\left(\dfrac{19}{2}\right)\sqrt{50}+\left(\dfrac{37}{2}\right)\sqrt{4}\right)\times\left(\dfrac{69}{5}\right)\\
&=&\left(\left(\left(-16\right)\sqrt{2}\right)+\dfrac{32}{9}-\left(\left(-\dfrac{130}{3}\right)\sqrt{2}\right)-4-\dfrac{27}{7}-3+\left(\dfrac{365}{7}\right)\sqrt{2}+\left(\left(\dfrac{34}{3}\right)\sqrt{2}\right)+\dfrac{8}{3}+\dfrac{47}{7}+\left(-\dfrac{245}{6}\right)\sqrt{2}+\left(\dfrac{95}{2}\right)\sqrt{2}+37\right)\times\left(\dfrac{69}{5}\right)\\
&=&\left(\left(\dfrac{2047}{21}\right)\sqrt{2}+\dfrac{2462}{63}\right)\left(\dfrac{69}{5}\right)\\
&=&\left(\dfrac{47081}{35}\right)\sqrt{2}+\dfrac{56626}{105}\\
&=&\left(\dfrac{47081}{35}\right)\sqrt{2}+\dfrac{56626}{105}\\
\end{eqnarray*}