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
Soit \( X=\left(\left(-\dfrac{50}{7}\right)\sqrt{4}+3+\left(5\right)\sqrt{8}+\left(-\dfrac{57}{7}\right)\sqrt{50}\right)-\left(-\dfrac{9}{4}-\left(\left(-8\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{18}+\left(-\dfrac{39}{2}\right)\sqrt{8}+\left(\dfrac{31}{7}\right)\sqrt{50}+\left(9\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{18}+\left(8\right)\sqrt{50}\right)\) et \( Y=\left(0\right)\sqrt{8}-\dfrac{49}{5}-\dfrac{61}{6}\) . 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{50}{7}\right)\sqrt{4}+3+\left(5\right)\sqrt{8}+\left(-\dfrac{57}{7}\right)\sqrt{50}\right)-\left(-\dfrac{9}{4}-\left(\left(-8\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{18}+\left(-\dfrac{39}{2}\right)\sqrt{8}+\left(\dfrac{31}{7}\right)\sqrt{50}+\left(9\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{18}+\left(8\right)\sqrt{50}\right)\right)+\left(\left(0\right)\sqrt{8}-\dfrac{49}{5}-\dfrac{61}{6}\right)\\
&=&\left(\left(-\dfrac{100}{7}+3+\left(10\right)\sqrt{2}+\left(-\dfrac{285}{7}\right)\sqrt{2}\right)-\left(-\dfrac{9}{4}-\left(\left(-40\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{95}{3}\right)\sqrt{2}+\left(\dfrac{117}{2}\right)\sqrt{2}+\left(-39\right)\sqrt{2}+\left(\dfrac{155}{7}\right)\sqrt{2}+\left(27\right)\sqrt{2}\right)-\left(\left(\dfrac{117}{2}\right)\sqrt{2}+\left(40\right)\sqrt{2}\right)\right)+\left(\left(0\right)\sqrt{2}-\dfrac{49}{5}-\dfrac{61}{6}\right)\\
&=&\left(-\dfrac{100}{7}+3+\left(10\right)\sqrt{2}+\left(-\dfrac{285}{7}\right)\sqrt{2}\right)-\left(-\dfrac{9}{4}-\left(\left(-40\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{95}{3}\right)\sqrt{2}+\left(\dfrac{117}{2}\right)\sqrt{2}+\left(-39\right)\sqrt{2}+\left(\dfrac{155}{7}\right)\sqrt{2}+\left(27\right)\sqrt{2}\right)-\left(\left(\dfrac{117}{2}\right)\sqrt{2}+\left(40\right)\sqrt{2}\right)+\left(0\right)\sqrt{2}-\dfrac{49}{5}-\dfrac{61}{6}\\
&=&-\dfrac{12181}{420}+\left(-\dfrac{5660}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{50}{7}\right)\sqrt{4}+3+\left(5\right)\sqrt{8}+\left(-\dfrac{57}{7}\right)\sqrt{50}\right)-\left(-\dfrac{9}{4}-\left(\left(-8\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{18}+\left(-\dfrac{39}{2}\right)\sqrt{8}+\left(\dfrac{31}{7}\right)\sqrt{50}+\left(9\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{18}+\left(8\right)\sqrt{50}\right)\right)-\left(\left(0\right)\sqrt{8}-\dfrac{49}{5}-\dfrac{61}{6}\right)\\
&=&\left(\left(-\dfrac{100}{7}+3+\left(10\right)\sqrt{2}+\left(-\dfrac{285}{7}\right)\sqrt{2}\right)-\left(-\dfrac{9}{4}-\left(\left(-40\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{95}{3}\right)\sqrt{2}+\left(\dfrac{117}{2}\right)\sqrt{2}+\left(-39\right)\sqrt{2}+\left(\dfrac{155}{7}\right)\sqrt{2}+\left(27\right)\sqrt{2}\right)-\left(\left(\dfrac{117}{2}\right)\sqrt{2}+\left(40\right)\sqrt{2}\right)\right)-\left(\left(0\right)\sqrt{2}-\dfrac{49}{5}-\dfrac{61}{6}\right)\\
&=&\left(-\dfrac{253}{28}+\left(-\dfrac{5660}{21}\right)\sqrt{2}\right)-\left(\left(0\right)\sqrt{2}-\dfrac{599}{30}\right)\\
&=&-\dfrac{253}{28}+\left(-\dfrac{5660}{21}\right)\sqrt{2}+\left(0\right)\sqrt{2}+\dfrac{599}{30}\\
&=&\dfrac{4591}{420}+\left(-\dfrac{5660}{21}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{50}{7}\right)\sqrt{4}+3+\left(5\right)\sqrt{8}+\left(-\dfrac{57}{7}\right)\sqrt{50}\right)-\left(-\dfrac{9}{4}-\left(\left(-8\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{50}+\left(\dfrac{39}{2}\right)\sqrt{18}+\left(-\dfrac{39}{2}\right)\sqrt{8}+\left(\dfrac{31}{7}\right)\sqrt{50}+\left(9\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{2}\right)\sqrt{18}+\left(8\right)\sqrt{50}\right)\right)\times\left(\left(0\right)\sqrt{8}-\dfrac{49}{5}-\dfrac{61}{6}\right)\\
&=&\left(\left(-\dfrac{100}{7}+3+\left(10\right)\sqrt{2}+\left(-\dfrac{285}{7}\right)\sqrt{2}\right)-\left(-\dfrac{9}{4}-\left(\left(-40\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{95}{3}\right)\sqrt{2}+\left(\dfrac{117}{2}\right)\sqrt{2}+\left(-39\right)\sqrt{2}+\left(\dfrac{155}{7}\right)\sqrt{2}+\left(27\right)\sqrt{2}\right)-\left(\left(\dfrac{117}{2}\right)\sqrt{2}+\left(40\right)\sqrt{2}\right)\right)\times\left(\left(0\right)\sqrt{2}-\dfrac{49}{5}-\dfrac{61}{6}\right)\\
&=&\left(-\dfrac{253}{28}+\left(-\dfrac{5660}{21}\right)\sqrt{2}\right)\left(\left(0\right)\sqrt{2}-\dfrac{599}{30}\right)\\
&=&\left(\dfrac{339034}{63}\right)\sqrt{2}+\dfrac{151547}{840}+\left(0\right)\sqrt{4}\\
&=&\left(\dfrac{339034}{63}\right)\sqrt{2}+\dfrac{151547}{840}\\
\end{eqnarray*}