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 \( X=\left(\left(-\dfrac{11}{3}\right)\sqrt{8}-7+\left(\dfrac{63}{5}\right)\sqrt{8}+\left(4\right)\sqrt{8}\right)-\left(\left(\left(\dfrac{62}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)\right)\) et \( Y=\left(\dfrac{1}{5}\right)\sqrt{18}\) . 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{11}{3}\right)\sqrt{8}-7+\left(\dfrac{63}{5}\right)\sqrt{8}+\left(4\right)\sqrt{8}\right)-\left(\left(\left(\dfrac{62}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)\right)\right)+\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{2}-7+\left(\dfrac{126}{5}\right)\sqrt{2}+\left(8\right)\sqrt{2}\right)-\left(\dfrac{124}{9}-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)\right)\right)+\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{22}{3}\right)\sqrt{2}-7+\left(\dfrac{126}{5}\right)\sqrt{2}+\left(8\right)\sqrt{2}\right)-\left(\dfrac{124}{9}-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)\right)+\left(\dfrac{3}{5}\right)\sqrt{2}\\
&=&\left(\dfrac{499}{5}\right)\sqrt{2}-\dfrac{187}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{11}{3}\right)\sqrt{8}-7+\left(\dfrac{63}{5}\right)\sqrt{8}+\left(4\right)\sqrt{8}\right)-\left(\left(\left(\dfrac{62}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)\right)\right)-\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{2}-7+\left(\dfrac{126}{5}\right)\sqrt{2}+\left(8\right)\sqrt{2}\right)-\left(\dfrac{124}{9}-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)\right)\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{496}{5}\right)\sqrt{2}-\dfrac{187}{9}\right)-\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{496}{5}\right)\sqrt{2}-\dfrac{187}{9}+\left(-\dfrac{3}{5}\right)\sqrt{2}\\
&=&\left(\dfrac{493}{5}\right)\sqrt{2}-\dfrac{187}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{11}{3}\right)\sqrt{8}-7+\left(\dfrac{63}{5}\right)\sqrt{8}+\left(4\right)\sqrt{8}\right)-\left(\left(\left(\dfrac{62}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{44}{3}\right)\sqrt{50}\right)\right)\right)\times\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)\\
&=&\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{2}-7+\left(\dfrac{126}{5}\right)\sqrt{2}+\left(8\right)\sqrt{2}\right)-\left(\dfrac{124}{9}-\left(\left(\dfrac{220}{3}\right)\sqrt{2}\right)\right)\right)\times\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{496}{5}\right)\sqrt{2}-\dfrac{187}{9}\right)\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{1488}{25}\right)\sqrt{4}+\left(-\dfrac{187}{15}\right)\sqrt{2}\\
&=&\dfrac{2976}{25}+\left(-\dfrac{187}{15}\right)\sqrt{2}\\
\end{eqnarray*}