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 \( X=-\dfrac{3}{7}+\left(\dfrac{67}{6}\right)\sqrt{45}+\left(-\dfrac{8}{5}\right)\sqrt{45}+\left(\dfrac{49}{3}\right)\sqrt{125}\) et \( Y=\dfrac{9}{5}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{3}{7}+\left(\dfrac{67}{6}\right)\sqrt{45}+\left(-\dfrac{8}{5}\right)\sqrt{45}+\left(\dfrac{49}{3}\right)\sqrt{125}\right)+\left(\dfrac{9}{5}\right)\\
&=&\left(-\dfrac{3}{7}+\left(\dfrac{67}{2}\right)\sqrt{5}+\left(-\dfrac{24}{5}\right)\sqrt{5}+\left(\dfrac{245}{3}\right)\sqrt{5}\right)+\left(\dfrac{9}{5}\right)\\
&=&-\dfrac{3}{7}+\left(\dfrac{67}{2}\right)\sqrt{5}+\left(-\dfrac{24}{5}\right)\sqrt{5}+\left(\dfrac{245}{3}\right)\sqrt{5}+\dfrac{9}{5}\\
&=&\dfrac{48}{35}+\left(\dfrac{3311}{30}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{3}{7}+\left(\dfrac{67}{6}\right)\sqrt{45}+\left(-\dfrac{8}{5}\right)\sqrt{45}+\left(\dfrac{49}{3}\right)\sqrt{125}\right)-\left(\dfrac{9}{5}\right)\\
&=&\left(-\dfrac{3}{7}+\left(\dfrac{67}{2}\right)\sqrt{5}+\left(-\dfrac{24}{5}\right)\sqrt{5}+\left(\dfrac{245}{3}\right)\sqrt{5}\right)-\left(\dfrac{9}{5}\right)\\
&=&\left(-\dfrac{3}{7}+\left(\dfrac{3311}{30}\right)\sqrt{5}\right)-\left(\dfrac{9}{5}\right)\\
&=&-\dfrac{3}{7}+\left(\dfrac{3311}{30}\right)\sqrt{5}+-\dfrac{9}{5}\\
&=&-\dfrac{78}{35}+\left(\dfrac{3311}{30}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{3}{7}+\left(\dfrac{67}{6}\right)\sqrt{45}+\left(-\dfrac{8}{5}\right)\sqrt{45}+\left(\dfrac{49}{3}\right)\sqrt{125}\right)\times\left(\dfrac{9}{5}\right)\\
&=&\left(-\dfrac{3}{7}+\left(\dfrac{67}{2}\right)\sqrt{5}+\left(-\dfrac{24}{5}\right)\sqrt{5}+\left(\dfrac{245}{3}\right)\sqrt{5}\right)\times\left(\dfrac{9}{5}\right)\\
&=&\left(-\dfrac{3}{7}+\left(\dfrac{3311}{30}\right)\sqrt{5}\right)\left(\dfrac{9}{5}\right)\\
&=&-\dfrac{27}{35}+\left(\dfrac{9933}{50}\right)\sqrt{5}\\
&=&-\dfrac{27}{35}+\left(\dfrac{9933}{50}\right)\sqrt{5}\\
\end{eqnarray*}