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 \( X=\left(\dfrac{2}{3}\right)\sqrt{125}\) et \( Y=\dfrac{14}{3}+\left(-\dfrac{19}{5}\right)\sqrt{20}+\left(-\dfrac{3}{2}\right)\sqrt{20}\) . 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{2}{3}\right)\sqrt{125}\right)+\left(\dfrac{14}{3}+\left(-\dfrac{19}{5}\right)\sqrt{20}+\left(-\dfrac{3}{2}\right)\sqrt{20}\right)\\
&=&\left(\left(\dfrac{10}{3}\right)\sqrt{5}\right)+\left(\dfrac{14}{3}+\left(-\dfrac{38}{5}\right)\sqrt{5}+\left(-3\right)\sqrt{5}\right)\\
&=&\left(\dfrac{10}{3}\right)\sqrt{5}+\dfrac{14}{3}+\left(-\dfrac{38}{5}\right)\sqrt{5}+\left(-3\right)\sqrt{5}\\
&=&\left(-\dfrac{109}{15}\right)\sqrt{5}+\dfrac{14}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{2}{3}\right)\sqrt{125}\right)-\left(\dfrac{14}{3}+\left(-\dfrac{19}{5}\right)\sqrt{20}+\left(-\dfrac{3}{2}\right)\sqrt{20}\right)\\
&=&\left(\left(\dfrac{10}{3}\right)\sqrt{5}\right)-\left(\dfrac{14}{3}+\left(-\dfrac{38}{5}\right)\sqrt{5}+\left(-3\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{10}{3}\right)\sqrt{5}\right)-\left(\dfrac{14}{3}+\left(-\dfrac{53}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{10}{3}\right)\sqrt{5}+-\dfrac{14}{3}+\left(\dfrac{53}{5}\right)\sqrt{5}\\
&=&\left(\dfrac{209}{15}\right)\sqrt{5}-\dfrac{14}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{2}{3}\right)\sqrt{125}\right)\times\left(\dfrac{14}{3}+\left(-\dfrac{19}{5}\right)\sqrt{20}+\left(-\dfrac{3}{2}\right)\sqrt{20}\right)\\
&=&\left(\left(\dfrac{10}{3}\right)\sqrt{5}\right)\times\left(\dfrac{14}{3}+\left(-\dfrac{38}{5}\right)\sqrt{5}+\left(-3\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{10}{3}\right)\sqrt{5}\right)\left(\dfrac{14}{3}+\left(-\dfrac{53}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{140}{9}\right)\sqrt{5}+\left(-\dfrac{106}{3}\right)\sqrt{25}\\
&=&\left(\dfrac{140}{9}\right)\sqrt{5}-\dfrac{530}{3}\\
\end{eqnarray*}