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 \( X=\left(-\dfrac{41}{2}-1+\left(\dfrac{29}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)\) et \( Y=\dfrac{46}{7}\) . 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{41}{2}-1+\left(\dfrac{29}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)\right)+\left(\dfrac{46}{7}\right)\\
&=&\left(\left(-\dfrac{41}{2}-1+\left(\dfrac{58}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{5}\right)+\dfrac{375}{8}\right)+\left(\dfrac{46}{7}\right)\\
&=&\left(-\dfrac{41}{2}-1+\left(\dfrac{58}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{5}\right)+\dfrac{375}{8}+\dfrac{46}{7}\\
&=&\dfrac{1789}{56}+\left(\dfrac{998}{63}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{41}{2}-1+\left(\dfrac{29}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)\right)-\left(\dfrac{46}{7}\right)\\
&=&\left(\left(-\dfrac{41}{2}-1+\left(\dfrac{58}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{5}\right)+\dfrac{375}{8}\right)-\left(\dfrac{46}{7}\right)\\
&=&\left(\dfrac{203}{8}+\left(\dfrac{998}{63}\right)\sqrt{5}\right)-\left(\dfrac{46}{7}\right)\\
&=&\dfrac{203}{8}+\left(\dfrac{998}{63}\right)\sqrt{5}+-\dfrac{46}{7}\\
&=&\dfrac{1053}{56}+\left(\dfrac{998}{63}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{41}{2}-1+\left(\dfrac{29}{7}\right)\sqrt{20}\right)-\left(\left(-\dfrac{34}{9}\right)\sqrt{20}\right)-\left(\left(-\dfrac{75}{8}\right)\sqrt{25}\right)\right)\times\left(\dfrac{46}{7}\right)\\
&=&\left(\left(-\dfrac{41}{2}-1+\left(\dfrac{58}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{5}\right)+\dfrac{375}{8}\right)\times\left(\dfrac{46}{7}\right)\\
&=&\left(\dfrac{203}{8}+\left(\dfrac{998}{63}\right)\sqrt{5}\right)\left(\dfrac{46}{7}\right)\\
&=&\dfrac{667}{4}+\left(\dfrac{45908}{441}\right)\sqrt{5}\\
&=&\dfrac{667}{4}+\left(\dfrac{45908}{441}\right)\sqrt{5}\\
\end{eqnarray*}