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 \( X=\left(\left(-\dfrac{33}{5}\right)\sqrt{4}+\left(-2\right)\sqrt{18}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)\) et \( Y=\left(-\dfrac{39}{4}\right)\sqrt{8}-\dfrac{41}{7}+\dfrac{66}{7}\) . 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{33}{5}\right)\sqrt{4}+\left(-2\right)\sqrt{18}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)\right)+\left(\left(-\dfrac{39}{4}\right)\sqrt{8}-\dfrac{41}{7}+\dfrac{66}{7}\right)\\
&=&\left(\left(-\dfrac{66}{5}+\left(-6\right)\sqrt{2}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)\right)+\left(\left(-\dfrac{39}{2}\right)\sqrt{2}-\dfrac{41}{7}+\dfrac{66}{7}\right)\\
&=&\left(-\dfrac{66}{5}+\left(-6\right)\sqrt{2}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)+\left(-\dfrac{39}{2}\right)\sqrt{2}-\dfrac{41}{7}+\dfrac{66}{7}\\
&=&-\dfrac{981}{280}+\left(-\dfrac{33}{14}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{33}{5}\right)\sqrt{4}+\left(-2\right)\sqrt{18}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{39}{4}\right)\sqrt{8}-\dfrac{41}{7}+\dfrac{66}{7}\right)\\
&=&\left(\left(-\dfrac{66}{5}+\left(-6\right)\sqrt{2}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{2}-\dfrac{41}{7}+\dfrac{66}{7}\right)\\
&=&\left(-\dfrac{283}{40}+\left(\dfrac{120}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{2}+\dfrac{25}{7}\right)\\
&=&-\dfrac{283}{40}+\left(\dfrac{120}{7}\right)\sqrt{2}+\left(\dfrac{39}{2}\right)\sqrt{2}-\dfrac{25}{7}\\
&=&-\dfrac{2981}{280}+\left(\dfrac{513}{14}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{33}{5}\right)\sqrt{4}+\left(-2\right)\sqrt{18}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)\right)\times\left(\left(-\dfrac{39}{4}\right)\sqrt{8}-\dfrac{41}{7}+\dfrac{66}{7}\right)\\
&=&\left(\left(-\dfrac{66}{5}+\left(-6\right)\sqrt{2}\right)+\dfrac{49}{8}-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)\right)\times\left(\left(-\dfrac{39}{2}\right)\sqrt{2}-\dfrac{41}{7}+\dfrac{66}{7}\right)\\
&=&\left(-\dfrac{283}{40}+\left(\dfrac{120}{7}\right)\sqrt{2}\right)\left(\left(-\dfrac{39}{2}\right)\sqrt{2}+\dfrac{25}{7}\right)\\
&=&\left(\dfrac{780813}{3920}\right)\sqrt{2}-\dfrac{1415}{56}+\left(-\dfrac{2340}{7}\right)\sqrt{4}\\
&=&\left(\dfrac{780813}{3920}\right)\sqrt{2}-\dfrac{38855}{56}\\
\end{eqnarray*}