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 \( X=\left(\dfrac{61}{6}\right)\sqrt{8}+\left(-2\right)\sqrt{8}+\left(\dfrac{73}{8}\right)\sqrt{50}\) et \( Y=\left(\left(-\dfrac{2}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{4}\right)\) . 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{61}{6}\right)\sqrt{8}+\left(-2\right)\sqrt{8}+\left(\dfrac{73}{8}\right)\sqrt{50}\right)+\left(\left(\left(-\dfrac{2}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{61}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{365}{8}\right)\sqrt{2}\right)+\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{2}\right)+\dfrac{47}{4}\right)\\
&=&\left(\dfrac{61}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{365}{8}\right)\sqrt{2}+\left(\left(-\dfrac{2}{3}\right)\sqrt{2}\right)+\dfrac{47}{4}\\
&=&\left(\dfrac{1471}{24}\right)\sqrt{2}+\dfrac{47}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{61}{6}\right)\sqrt{8}+\left(-2\right)\sqrt{8}+\left(\dfrac{73}{8}\right)\sqrt{50}\right)-\left(\left(\left(-\dfrac{2}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{61}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{365}{8}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{2}\right)+\dfrac{47}{4}\right)\\
&=&\left(\left(\dfrac{1487}{24}\right)\sqrt{2}\right)-\left(\left(-\dfrac{2}{3}\right)\sqrt{2}+\dfrac{47}{4}\right)\\
&=&\left(\dfrac{1487}{24}\right)\sqrt{2}+\left(\dfrac{2}{3}\right)\sqrt{2}-\dfrac{47}{4}\\
&=&\left(\dfrac{501}{8}\right)\sqrt{2}-\dfrac{47}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{61}{6}\right)\sqrt{8}+\left(-2\right)\sqrt{8}+\left(\dfrac{73}{8}\right)\sqrt{50}\right)\times\left(\left(\left(-\dfrac{2}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{47}{8}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\dfrac{61}{3}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{365}{8}\right)\sqrt{2}\right)\times\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{2}\right)+\dfrac{47}{4}\right)\\
&=&\left(\left(\dfrac{1487}{24}\right)\sqrt{2}\right)\left(\left(-\dfrac{2}{3}\right)\sqrt{2}+\dfrac{47}{4}\right)\\
&=&\left(-\dfrac{1487}{36}\right)\sqrt{4}+\left(\dfrac{69889}{96}\right)\sqrt{2}\\
&=&-\dfrac{1487}{18}+\left(\dfrac{69889}{96}\right)\sqrt{2}\\
\end{eqnarray*}