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 \( X=\left(\left(6\right)\sqrt{50}-\dfrac{79}{6}+\left(\dfrac{51}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{50}\right)\) et \( Y=\left(\dfrac{57}{7}\right)\sqrt{8}\) . 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(6\right)\sqrt{50}-\dfrac{79}{6}+\left(\dfrac{51}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{50}\right)\right)+\left(\left(\dfrac{57}{7}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(30\right)\sqrt{2}-\dfrac{79}{6}+\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{95}{9}\right)\sqrt{2}\right)\right)+\left(\left(\dfrac{114}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(30\right)\sqrt{2}-\dfrac{79}{6}+\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{95}{9}\right)\sqrt{2}\right)+\left(\dfrac{114}{7}\right)\sqrt{2}\\
&=&\left(\dfrac{17681}{315}\right)\sqrt{2}-\dfrac{79}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(6\right)\sqrt{50}-\dfrac{79}{6}+\left(\dfrac{51}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{57}{7}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(30\right)\sqrt{2}-\dfrac{79}{6}+\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{95}{9}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{114}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{1793}{45}\right)\sqrt{2}-\dfrac{79}{6}\right)-\left(\left(\dfrac{114}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{1793}{45}\right)\sqrt{2}-\dfrac{79}{6}+\left(-\dfrac{114}{7}\right)\sqrt{2}\\
&=&\left(\dfrac{7421}{315}\right)\sqrt{2}-\dfrac{79}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(6\right)\sqrt{50}-\dfrac{79}{6}+\left(\dfrac{51}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{19}{9}\right)\sqrt{50}\right)\right)\times\left(\left(\dfrac{57}{7}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(30\right)\sqrt{2}-\dfrac{79}{6}+\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{95}{9}\right)\sqrt{2}\right)\right)\times\left(\left(\dfrac{114}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{1793}{45}\right)\sqrt{2}-\dfrac{79}{6}\right)\left(\left(\dfrac{114}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{68134}{105}\right)\sqrt{4}+\left(-\dfrac{1501}{7}\right)\sqrt{2}\\
&=&\dfrac{136268}{105}+\left(-\dfrac{1501}{7}\right)\sqrt{2}\\
\end{eqnarray*}