L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{2}{7}\right)\sqrt{25}+9+\left(\dfrac{29}{3}\right)\sqrt{25}+\dfrac{46}{7}-\left(\left(\dfrac{68}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\) et \( Y=-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{4}\right)\sqrt{20}\right)+\left(4\right)\sqrt{45}+\left(3\right)\sqrt{125}+\left(\left(\dfrac{25}{7}\right)\sqrt{125}\right)-5-\left(\left(-\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{25}\right)\) . 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}{7}\right)\sqrt{25}+9+\left(\dfrac{29}{3}\right)\sqrt{25}+\dfrac{46}{7}-\left(\left(\dfrac{68}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\right)+\left(-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{4}\right)\sqrt{20}\right)+\left(4\right)\sqrt{45}+\left(3\right)\sqrt{125}+\left(\left(\dfrac{25}{7}\right)\sqrt{125}\right)-5-\left(\left(-\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{10}{7}+9+\dfrac{145}{3}+\dfrac{46}{7}-\left(\left(\dfrac{204}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\right)+\left(-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{2}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(\left(\dfrac{125}{7}\right)\sqrt{5}\right)-5-\left(\left(-\dfrac{3}{4}\right)\sqrt{5}\right)+17\right)\\
&=&\dfrac{10}{7}+9+\dfrac{145}{3}+\dfrac{46}{7}-\left(\left(\dfrac{204}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{2}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(\left(\dfrac{125}{7}\right)\sqrt{5}\right)-5-\left(\left(-\dfrac{3}{4}\right)\sqrt{5}\right)+17\\
&=&\dfrac{11839}{210}+\left(-\dfrac{1921}{70}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{2}{7}\right)\sqrt{25}+9+\left(\dfrac{29}{3}\right)\sqrt{25}+\dfrac{46}{7}-\left(\left(\dfrac{68}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\right)-\left(-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{4}\right)\sqrt{20}\right)+\left(4\right)\sqrt{45}+\left(3\right)\sqrt{125}+\left(\left(\dfrac{25}{7}\right)\sqrt{125}\right)-5-\left(\left(-\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{10}{7}+9+\dfrac{145}{3}+\dfrac{46}{7}-\left(\left(\dfrac{204}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\right)-\left(-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{2}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(\left(\dfrac{125}{7}\right)\sqrt{5}\right)-5-\left(\left(-\dfrac{3}{4}\right)\sqrt{5}\right)+17\right)\\
&=&\left(\dfrac{2981}{42}+\left(-\dfrac{1591}{20}\right)\sqrt{5}\right)-\left(-\dfrac{73}{5}+\left(\dfrac{1459}{28}\right)\sqrt{5}\right)\\
&=&\dfrac{2981}{42}+\left(-\dfrac{1591}{20}\right)\sqrt{5}+\dfrac{73}{5}+\left(-\dfrac{1459}{28}\right)\sqrt{5}\\
&=&\dfrac{17971}{210}+\left(-\dfrac{4608}{35}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{2}{7}\right)\sqrt{25}+9+\left(\dfrac{29}{3}\right)\sqrt{25}+\dfrac{46}{7}-\left(\left(\dfrac{68}{5}\right)\sqrt{45}\right)-\left(\left(\dfrac{31}{4}\right)\sqrt{125}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\right)\times\left(-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{4}\right)\sqrt{20}\right)+\left(4\right)\sqrt{45}+\left(3\right)\sqrt{125}+\left(\left(\dfrac{25}{7}\right)\sqrt{125}\right)-5-\left(\left(-\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\left(-\dfrac{17}{5}\right)\sqrt{25}\right)\right)\\
&=&\left(\dfrac{10}{7}+9+\dfrac{145}{3}+\dfrac{46}{7}-\left(\left(\dfrac{204}{5}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{4}\right)\sqrt{5}\right)+\dfrac{50}{7}-\dfrac{3}{2}+\dfrac{55}{8}-\dfrac{55}{8}\right)\times\left(-\dfrac{68}{5}-8-5-\left(\left(-\dfrac{13}{2}\right)\sqrt{5}\right)+\left(12\right)\sqrt{5}+\left(15\right)\sqrt{5}+\left(\left(\dfrac{125}{7}\right)\sqrt{5}\right)-5-\left(\left(-\dfrac{3}{4}\right)\sqrt{5}\right)+17\right)\\
&=&\left(\dfrac{2981}{42}+\left(-\dfrac{1591}{20}\right)\sqrt{5}\right)\left(-\dfrac{73}{5}+\left(\dfrac{1459}{28}\right)\sqrt{5}\right)\\
&=&-\dfrac{217613}{210}+\left(\dfrac{142878017}{29400}\right)\sqrt{5}+\left(-\dfrac{2321269}{560}\right)\sqrt{25}\\
&=&-\dfrac{36559939}{1680}+\left(\dfrac{142878017}{29400}\right)\sqrt{5}\\
\end{eqnarray*}