L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-9-\left(\left(-5\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}\right)+\left(\left(-\dfrac{59}{2}\right)\sqrt{45}\right)-8-\left(\left(-\dfrac{77}{5}\right)\sqrt{45}\right)\) et \( Y=\left(\left(\dfrac{14}{9}\right)\sqrt{20}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)+\left(\left(\dfrac{73}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{125}\right)-\left(\left(-8\right)\sqrt{20}\right)+\dfrac{17}{3}-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{45}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-9-\left(\left(-5\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}\right)+\left(\left(-\dfrac{59}{2}\right)\sqrt{45}\right)-8-\left(\left(-\dfrac{77}{5}\right)\sqrt{45}\right)\right)+\left(\left(\left(\dfrac{14}{9}\right)\sqrt{20}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)+\left(\left(\dfrac{73}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{125}\right)-\left(\left(-8\right)\sqrt{20}\right)+\dfrac{17}{3}-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{45}\right)\right)\\
&=&\left(-9-\left(\left(-15\right)\sqrt{5}\right)-0-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{177}{2}\right)\sqrt{5}\right)-8-\left(\left(-\dfrac{231}{5}\right)\sqrt{5}\right)\right)+\left(\left(\left(\dfrac{28}{9}\right)\sqrt{5}\right)-\dfrac{85}{3}+\left(\left(\dfrac{365}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{275}{7}\right)\sqrt{5}\right)-\left(\left(-16\right)\sqrt{5}\right)+\dfrac{17}{3}-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{183}{7}\right)\sqrt{5}\right)\right)\\
&=&-9-\left(\left(-15\right)\sqrt{5}\right)-0-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{177}{2}\right)\sqrt{5}\right)-8-\left(\left(-\dfrac{231}{5}\right)\sqrt{5}\right)+\left(\left(\dfrac{28}{9}\right)\sqrt{5}\right)-\dfrac{85}{3}+\left(\left(\dfrac{365}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{275}{7}\right)\sqrt{5}\right)-\left(\left(-16\right)\sqrt{5}\right)+\dfrac{17}{3}-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{183}{7}\right)\sqrt{5}\right)\\
&=&-\dfrac{119}{3}+\left(\dfrac{367639}{2520}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-9-\left(\left(-5\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}\right)+\left(\left(-\dfrac{59}{2}\right)\sqrt{45}\right)-8-\left(\left(-\dfrac{77}{5}\right)\sqrt{45}\right)\right)-\left(\left(\left(\dfrac{14}{9}\right)\sqrt{20}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)+\left(\left(\dfrac{73}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{125}\right)-\left(\left(-8\right)\sqrt{20}\right)+\dfrac{17}{3}-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{45}\right)\right)\\
&=&\left(-9-\left(\left(-15\right)\sqrt{5}\right)-0-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{177}{2}\right)\sqrt{5}\right)-8-\left(\left(-\dfrac{231}{5}\right)\sqrt{5}\right)\right)-\left(\left(\left(\dfrac{28}{9}\right)\sqrt{5}\right)-\dfrac{85}{3}+\left(\left(\dfrac{365}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{275}{7}\right)\sqrt{5}\right)-\left(\left(-16\right)\sqrt{5}\right)+\dfrac{17}{3}-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{183}{7}\right)\sqrt{5}\right)\right)\\
&=&\left(-17+\left(\dfrac{833}{40}\right)\sqrt{5}\right)-\left(\left(\dfrac{7879}{63}\right)\sqrt{5}-\dfrac{68}{3}\right)\\
&=&-17+\left(\dfrac{833}{40}\right)\sqrt{5}+\left(-\dfrac{7879}{63}\right)\sqrt{5}+\dfrac{68}{3}\\
&=&\dfrac{17}{3}+\left(-\dfrac{262681}{2520}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-9-\left(\left(-5\right)\sqrt{45}\right)-\left(\left(0\right)\sqrt{25}\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}\right)+\left(\left(-\dfrac{59}{2}\right)\sqrt{45}\right)-8-\left(\left(-\dfrac{77}{5}\right)\sqrt{45}\right)\right)\times\left(\left(\left(\dfrac{14}{9}\right)\sqrt{20}\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)+\left(\left(\dfrac{73}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{55}{7}\right)\sqrt{125}\right)-\left(\left(-8\right)\sqrt{20}\right)+\dfrac{17}{3}-\left(\left(\dfrac{6}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{7}\right)\sqrt{45}\right)\right)\\
&=&\left(-9-\left(\left(-15\right)\sqrt{5}\right)-0-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}\right)+\left(\left(-\dfrac{177}{2}\right)\sqrt{5}\right)-8-\left(\left(-\dfrac{231}{5}\right)\sqrt{5}\right)\right)\times\left(\left(\left(\dfrac{28}{9}\right)\sqrt{5}\right)-\dfrac{85}{3}+\left(\left(\dfrac{365}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{275}{7}\right)\sqrt{5}\right)-\left(\left(-16\right)\sqrt{5}\right)+\dfrac{17}{3}-\left(\left(\dfrac{18}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{183}{7}\right)\sqrt{5}\right)\right)\\
&=&\left(-17+\left(\dfrac{833}{40}\right)\sqrt{5}\right)\left(\left(\dfrac{7879}{63}\right)\sqrt{5}-\dfrac{68}{3}\right)\\
&=&\left(-\dfrac{1636811}{630}\right)\sqrt{5}+\dfrac{1156}{3}+\left(\dfrac{937601}{360}\right)\sqrt{25}\\
&=&\left(-\dfrac{1636811}{630}\right)\sqrt{5}+\dfrac{965345}{72}\\
\end{eqnarray*}