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 \( X=\left(5\right)\sqrt{20}\) et \( Y=\dfrac{52}{3}+\left(-\dfrac{41}{7}\right)\sqrt{20}+\left(-\dfrac{65}{7}\right)\sqrt{25}+\left(\left(\dfrac{27}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{25}\right)-\dfrac{67}{4}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(5\right)\sqrt{20}\right)+\left(\dfrac{52}{3}+\left(-\dfrac{41}{7}\right)\sqrt{20}+\left(-\dfrac{65}{7}\right)\sqrt{25}+\left(\left(\dfrac{27}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{25}\right)-\dfrac{67}{4}\right)\\
&=&\left(\left(10\right)\sqrt{5}\right)+\left(\dfrac{52}{3}+\left(-\dfrac{82}{7}\right)\sqrt{5}-\dfrac{325}{7}+\dfrac{135}{4}+\dfrac{25}{4}-\dfrac{67}{4}\right)\\
&=&\left(10\right)\sqrt{5}+\dfrac{52}{3}+\left(-\dfrac{82}{7}\right)\sqrt{5}-\dfrac{325}{7}+\dfrac{135}{4}+\dfrac{25}{4}-\dfrac{67}{4}\\
&=&\left(-\dfrac{12}{7}\right)\sqrt{5}-\dfrac{491}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(5\right)\sqrt{20}\right)-\left(\dfrac{52}{3}+\left(-\dfrac{41}{7}\right)\sqrt{20}+\left(-\dfrac{65}{7}\right)\sqrt{25}+\left(\left(\dfrac{27}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{25}\right)-\dfrac{67}{4}\right)\\
&=&\left(\left(10\right)\sqrt{5}\right)-\left(\dfrac{52}{3}+\left(-\dfrac{82}{7}\right)\sqrt{5}-\dfrac{325}{7}+\dfrac{135}{4}+\dfrac{25}{4}-\dfrac{67}{4}\right)\\
&=&\left(\left(10\right)\sqrt{5}\right)-\left(-\dfrac{491}{84}+\left(-\dfrac{82}{7}\right)\sqrt{5}\right)\\
&=&\left(10\right)\sqrt{5}+\dfrac{491}{84}+\left(\dfrac{82}{7}\right)\sqrt{5}\\
&=&\left(\dfrac{152}{7}\right)\sqrt{5}+\dfrac{491}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(5\right)\sqrt{20}\right)\times\left(\dfrac{52}{3}+\left(-\dfrac{41}{7}\right)\sqrt{20}+\left(-\dfrac{65}{7}\right)\sqrt{25}+\left(\left(\dfrac{27}{4}\right)\sqrt{25}\right)-\left(\left(-\dfrac{5}{4}\right)\sqrt{25}\right)-\dfrac{67}{4}\right)\\
&=&\left(\left(10\right)\sqrt{5}\right)\times\left(\dfrac{52}{3}+\left(-\dfrac{82}{7}\right)\sqrt{5}-\dfrac{325}{7}+\dfrac{135}{4}+\dfrac{25}{4}-\dfrac{67}{4}\right)\\
&=&\left(\left(10\right)\sqrt{5}\right)\left(-\dfrac{491}{84}+\left(-\dfrac{82}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{2455}{42}\right)\sqrt{5}+\left(-\dfrac{820}{7}\right)\sqrt{25}\\
&=&\left(-\dfrac{2455}{42}\right)\sqrt{5}-\dfrac{4100}{7}\\
\end{eqnarray*}