L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(3\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\dfrac{49}{2}\right)\sqrt{8}+\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{15}{7}\right)\sqrt{4}\right)-\dfrac{17}{5}\) et \( Y=\left(\left(\dfrac{44}{3}\right)\sqrt{8}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{28}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{73}{5}\right)\sqrt{50}+\left(-\dfrac{16}{7}\right)\sqrt{18}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(3\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\dfrac{49}{2}\right)\sqrt{8}+\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{15}{7}\right)\sqrt{4}\right)-\dfrac{17}{5}\right)+\left(\left(\left(\dfrac{44}{3}\right)\sqrt{8}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{28}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{73}{5}\right)\sqrt{50}+\left(-\dfrac{16}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(15\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(49\right)\sqrt{2}+\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{2}\right)-\dfrac{30}{7}-\dfrac{17}{5}\right)+\left(\left(\left(\dfrac{88}{3}\right)\sqrt{2}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{140}{3}\right)\sqrt{2}+39\right)-\left(\left(-73\right)\sqrt{2}+\left(-\dfrac{48}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(15\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(49\right)\sqrt{2}+\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{2}\right)-\dfrac{30}{7}-\dfrac{17}{5}+\left(\left(\dfrac{88}{3}\right)\sqrt{2}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{140}{3}\right)\sqrt{2}+39\right)-\left(\left(-73\right)\sqrt{2}+\left(-\dfrac{48}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{11533}{168}\right)\sqrt{2}-\dfrac{11869}{210}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(3\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\dfrac{49}{2}\right)\sqrt{8}+\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{15}{7}\right)\sqrt{4}\right)-\dfrac{17}{5}\right)-\left(\left(\left(\dfrac{44}{3}\right)\sqrt{8}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{28}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{73}{5}\right)\sqrt{50}+\left(-\dfrac{16}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(15\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(49\right)\sqrt{2}+\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{2}\right)-\dfrac{30}{7}-\dfrac{17}{5}\right)-\left(\left(\left(\dfrac{88}{3}\right)\sqrt{2}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{140}{3}\right)\sqrt{2}+39\right)-\left(\left(-73\right)\sqrt{2}+\left(-\dfrac{48}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{2093}{24}\right)\sqrt{2}-\dfrac{269}{35}\right)-\left(\left(\dfrac{1091}{7}\right)\sqrt{2}-\dfrac{293}{6}\right)\\
&=&\left(-\dfrac{2093}{24}\right)\sqrt{2}-\dfrac{269}{35}+\left(-\dfrac{1091}{7}\right)\sqrt{2}+\dfrac{293}{6}\\
&=&\left(-\dfrac{40835}{168}\right)\sqrt{2}+\dfrac{8641}{210}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(3\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(-\dfrac{35}{2}\right)\sqrt{50}+\left(\dfrac{49}{2}\right)\sqrt{8}+\left(\left(\dfrac{25}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{19}{8}\right)\sqrt{18}\right)-\left(\left(\dfrac{15}{7}\right)\sqrt{4}\right)-\dfrac{17}{5}\right)\times\left(\left(\left(\dfrac{44}{3}\right)\sqrt{8}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{28}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{4}\right)-\left(\left(-\dfrac{73}{5}\right)\sqrt{50}+\left(-\dfrac{16}{7}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(15\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(-\dfrac{175}{2}\right)\sqrt{2}+\left(49\right)\sqrt{2}+\left(\left(\dfrac{50}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{57}{8}\right)\sqrt{2}\right)-\dfrac{30}{7}-\dfrac{17}{5}\right)\times\left(\left(\left(\dfrac{88}{3}\right)\sqrt{2}\right)-\dfrac{59}{6}-\left(\left(-\dfrac{140}{3}\right)\sqrt{2}+39\right)-\left(\left(-73\right)\sqrt{2}+\left(-\dfrac{48}{7}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{2093}{24}\right)\sqrt{2}-\dfrac{269}{35}\right)\left(\left(\dfrac{1091}{7}\right)\sqrt{2}-\dfrac{293}{6}\right)\\
&=&\left(-\dfrac{326209}{24}\right)\sqrt{4}+\left(\dfrac{107985029}{35280}\right)\sqrt{2}+\dfrac{78817}{210}\\
&=&-\dfrac{3753227}{140}+\left(\dfrac{107985029}{35280}\right)\sqrt{2}\\
\end{eqnarray*}