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 \( X=\left(\dfrac{25}{6}\right)\sqrt{8}\) et \( Y=\left(-\dfrac{37}{2}\right)\sqrt{4}+\left(\dfrac{14}{5}\right)\sqrt{8}+\left(-\dfrac{1}{5}\right)\sqrt{50}+\left(-\dfrac{23}{9}\right)\sqrt{50}+\left(\dfrac{35}{6}\right)\sqrt{18}+\left(-\dfrac{36}{7}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{25}{6}\right)\sqrt{8}\right)+\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\left(\dfrac{14}{5}\right)\sqrt{8}+\left(-\dfrac{1}{5}\right)\sqrt{50}+\left(-\dfrac{23}{9}\right)\sqrt{50}+\left(\dfrac{35}{6}\right)\sqrt{18}+\left(-\dfrac{36}{7}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)+\left(-37+\left(\dfrac{28}{5}\right)\sqrt{2}+\left(-1\right)\sqrt{2}+\left(-\dfrac{115}{9}\right)\sqrt{2}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{180}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{25}{3}\right)\sqrt{2}-37+\left(\dfrac{28}{5}\right)\sqrt{2}+\left(-1\right)\sqrt{2}+\left(-\dfrac{115}{9}\right)\sqrt{2}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{180}{7}\right)\sqrt{2}\\
&=&\left(-\dfrac{5077}{630}\right)\sqrt{2}-37\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{25}{6}\right)\sqrt{8}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\left(\dfrac{14}{5}\right)\sqrt{8}+\left(-\dfrac{1}{5}\right)\sqrt{50}+\left(-\dfrac{23}{9}\right)\sqrt{50}+\left(\dfrac{35}{6}\right)\sqrt{18}+\left(-\dfrac{36}{7}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)-\left(-37+\left(\dfrac{28}{5}\right)\sqrt{2}+\left(-1\right)\sqrt{2}+\left(-\dfrac{115}{9}\right)\sqrt{2}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{180}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)-\left(-37+\left(-\dfrac{10327}{630}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{25}{3}\right)\sqrt{2}+37+\left(\dfrac{10327}{630}\right)\sqrt{2}\\
&=&\left(\dfrac{15577}{630}\right)\sqrt{2}+37\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{25}{6}\right)\sqrt{8}\right)\times\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\left(\dfrac{14}{5}\right)\sqrt{8}+\left(-\dfrac{1}{5}\right)\sqrt{50}+\left(-\dfrac{23}{9}\right)\sqrt{50}+\left(\dfrac{35}{6}\right)\sqrt{18}+\left(-\dfrac{36}{7}\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\times\left(-37+\left(\dfrac{28}{5}\right)\sqrt{2}+\left(-1\right)\sqrt{2}+\left(-\dfrac{115}{9}\right)\sqrt{2}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{180}{7}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\left(-37+\left(-\dfrac{10327}{630}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{925}{3}\right)\sqrt{2}+\left(-\dfrac{51635}{378}\right)\sqrt{4}\\
&=&\left(-\dfrac{925}{3}\right)\sqrt{2}-\dfrac{51635}{189}\\
\end{eqnarray*}