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 \( X=\left(-\dfrac{62}{9}\right)\sqrt{18}+\left(-2\right)\sqrt{18}+\left(-\dfrac{19}{6}\right)\sqrt{4}+\left(-\dfrac{58}{9}\right)\sqrt{50}+\left(-\dfrac{7}{2}\right)\sqrt{4}\) et \( Y=\left(\left(1\right)\sqrt{8}\right)-\left(\left(3\right)\sqrt{4}\right)-\left(\left(\dfrac{39}{5}\right)\sqrt{8}\right)+\left(-\dfrac{71}{8}\right)\sqrt{8}+\dfrac{45}{8}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{62}{9}\right)\sqrt{18}+\left(-2\right)\sqrt{18}+\left(-\dfrac{19}{6}\right)\sqrt{4}+\left(-\dfrac{58}{9}\right)\sqrt{50}+\left(-\dfrac{7}{2}\right)\sqrt{4}\right)+\left(\left(\left(1\right)\sqrt{8}\right)-\left(\left(3\right)\sqrt{4}\right)-\left(\left(\dfrac{39}{5}\right)\sqrt{8}\right)+\left(-\dfrac{71}{8}\right)\sqrt{8}+\dfrac{45}{8}\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}+\left(-6\right)\sqrt{2}-\dfrac{19}{3}+\left(-\dfrac{290}{9}\right)\sqrt{2}-7\right)+\left(\left(\left(2\right)\sqrt{2}\right)-6-\left(\left(\dfrac{78}{5}\right)\sqrt{2}\right)+\left(-\dfrac{71}{4}\right)\sqrt{2}+\dfrac{45}{8}\right)\\
&=&\left(-\dfrac{62}{3}\right)\sqrt{2}+\left(-6\right)\sqrt{2}-\dfrac{19}{3}+\left(-\dfrac{290}{9}\right)\sqrt{2}-7+\left(\left(2\right)\sqrt{2}\right)-6-\left(\left(\dfrac{78}{5}\right)\sqrt{2}\right)+\left(-\dfrac{71}{4}\right)\sqrt{2}+\dfrac{45}{8}\\
&=&\left(-\dfrac{16243}{180}\right)\sqrt{2}-\dfrac{329}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{62}{9}\right)\sqrt{18}+\left(-2\right)\sqrt{18}+\left(-\dfrac{19}{6}\right)\sqrt{4}+\left(-\dfrac{58}{9}\right)\sqrt{50}+\left(-\dfrac{7}{2}\right)\sqrt{4}\right)-\left(\left(\left(1\right)\sqrt{8}\right)-\left(\left(3\right)\sqrt{4}\right)-\left(\left(\dfrac{39}{5}\right)\sqrt{8}\right)+\left(-\dfrac{71}{8}\right)\sqrt{8}+\dfrac{45}{8}\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}+\left(-6\right)\sqrt{2}-\dfrac{19}{3}+\left(-\dfrac{290}{9}\right)\sqrt{2}-7\right)-\left(\left(\left(2\right)\sqrt{2}\right)-6-\left(\left(\dfrac{78}{5}\right)\sqrt{2}\right)+\left(-\dfrac{71}{4}\right)\sqrt{2}+\dfrac{45}{8}\right)\\
&=&\left(\left(-\dfrac{530}{9}\right)\sqrt{2}-\dfrac{40}{3}\right)-\left(\left(-\dfrac{627}{20}\right)\sqrt{2}-\dfrac{3}{8}\right)\\
&=&\left(-\dfrac{530}{9}\right)\sqrt{2}-\dfrac{40}{3}+\left(\dfrac{627}{20}\right)\sqrt{2}+\dfrac{3}{8}\\
&=&\left(-\dfrac{4957}{180}\right)\sqrt{2}-\dfrac{311}{24}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{62}{9}\right)\sqrt{18}+\left(-2\right)\sqrt{18}+\left(-\dfrac{19}{6}\right)\sqrt{4}+\left(-\dfrac{58}{9}\right)\sqrt{50}+\left(-\dfrac{7}{2}\right)\sqrt{4}\right)\times\left(\left(\left(1\right)\sqrt{8}\right)-\left(\left(3\right)\sqrt{4}\right)-\left(\left(\dfrac{39}{5}\right)\sqrt{8}\right)+\left(-\dfrac{71}{8}\right)\sqrt{8}+\dfrac{45}{8}\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}+\left(-6\right)\sqrt{2}-\dfrac{19}{3}+\left(-\dfrac{290}{9}\right)\sqrt{2}-7\right)\times\left(\left(\left(2\right)\sqrt{2}\right)-6-\left(\left(\dfrac{78}{5}\right)\sqrt{2}\right)+\left(-\dfrac{71}{4}\right)\sqrt{2}+\dfrac{45}{8}\right)\\
&=&\left(\left(-\dfrac{530}{9}\right)\sqrt{2}-\dfrac{40}{3}\right)\left(\left(-\dfrac{627}{20}\right)\sqrt{2}-\dfrac{3}{8}\right)\\
&=&\left(\dfrac{11077}{6}\right)\sqrt{4}+\left(\dfrac{5281}{12}\right)\sqrt{2}+5\\
&=&\dfrac{11092}{3}+\left(\dfrac{5281}{12}\right)\sqrt{2}\\
\end{eqnarray*}