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 \( X=\left(-\dfrac{29}{5}\right)\sqrt{45}+\left(\left(-\dfrac{77}{3}\right)\sqrt{25}\right)+\dfrac{64}{7}-\left(\left(-\dfrac{32}{3}\right)\sqrt{20}\right)+\left(\left(-\dfrac{5}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{125}\right)-7\) et \( Y=7+\left(-\dfrac{52}{7}\right)\sqrt{45}\) . 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{29}{5}\right)\sqrt{45}+\left(\left(-\dfrac{77}{3}\right)\sqrt{25}\right)+\dfrac{64}{7}-\left(\left(-\dfrac{32}{3}\right)\sqrt{20}\right)+\left(\left(-\dfrac{5}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{125}\right)-7\right)+\left(7+\left(-\dfrac{52}{7}\right)\sqrt{45}\right)\\
&=&\left(\left(-\dfrac{87}{5}\right)\sqrt{5}-\dfrac{385}{3}+\dfrac{64}{7}-\left(\left(-\dfrac{64}{3}\right)\sqrt{5}\right)-\dfrac{25}{3}-\left(\left(\dfrac{55}{4}\right)\sqrt{5}\right)-7\right)+\left(7+\left(-\dfrac{156}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{87}{5}\right)\sqrt{5}-\dfrac{385}{3}+\dfrac{64}{7}-\left(\left(-\dfrac{64}{3}\right)\sqrt{5}\right)-\dfrac{25}{3}-\left(\left(\dfrac{55}{4}\right)\sqrt{5}\right)-7+7+\left(-\dfrac{156}{7}\right)\sqrt{5}\\
&=&\left(-\dfrac{13483}{420}\right)\sqrt{5}-\dfrac{2678}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{29}{5}\right)\sqrt{45}+\left(\left(-\dfrac{77}{3}\right)\sqrt{25}\right)+\dfrac{64}{7}-\left(\left(-\dfrac{32}{3}\right)\sqrt{20}\right)+\left(\left(-\dfrac{5}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{125}\right)-7\right)-\left(7+\left(-\dfrac{52}{7}\right)\sqrt{45}\right)\\
&=&\left(\left(-\dfrac{87}{5}\right)\sqrt{5}-\dfrac{385}{3}+\dfrac{64}{7}-\left(\left(-\dfrac{64}{3}\right)\sqrt{5}\right)-\dfrac{25}{3}-\left(\left(\dfrac{55}{4}\right)\sqrt{5}\right)-7\right)-\left(7+\left(-\dfrac{156}{7}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{589}{60}\right)\sqrt{5}-\dfrac{2825}{21}\right)-\left(7+\left(-\dfrac{156}{7}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{589}{60}\right)\sqrt{5}-\dfrac{2825}{21}+-7+\left(\dfrac{156}{7}\right)\sqrt{5}\\
&=&\left(\dfrac{5237}{420}\right)\sqrt{5}-\dfrac{2972}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{29}{5}\right)\sqrt{45}+\left(\left(-\dfrac{77}{3}\right)\sqrt{25}\right)+\dfrac{64}{7}-\left(\left(-\dfrac{32}{3}\right)\sqrt{20}\right)+\left(\left(-\dfrac{5}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{125}\right)-7\right)\times\left(7+\left(-\dfrac{52}{7}\right)\sqrt{45}\right)\\
&=&\left(\left(-\dfrac{87}{5}\right)\sqrt{5}-\dfrac{385}{3}+\dfrac{64}{7}-\left(\left(-\dfrac{64}{3}\right)\sqrt{5}\right)-\dfrac{25}{3}-\left(\left(\dfrac{55}{4}\right)\sqrt{5}\right)-7\right)\times\left(7+\left(-\dfrac{156}{7}\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{589}{60}\right)\sqrt{5}-\dfrac{2825}{21}\right)\left(7+\left(-\dfrac{156}{7}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{8611973}{2940}\right)\sqrt{5}+\left(\dfrac{7657}{35}\right)\sqrt{25}-\dfrac{2825}{3}\\
&=&\left(\dfrac{8611973}{2940}\right)\sqrt{5}+\dfrac{3196}{21}\\
\end{eqnarray*}