L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=0+\left(-\dfrac{3}{2}\right)\sqrt{125}+3+\dfrac{41}{4}+\dfrac{6}{5}+\left(\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{77}{6}\right)\sqrt{25}+\dfrac{41}{4}\) et \( Y=\left(\left(2\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{45}\right)+\left(\dfrac{23}{7}\right)\sqrt{125}-\dfrac{51}{5}-\left(\left(\dfrac{29}{2}\right)\sqrt{45}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{59}{2}\right)\sqrt{125}\right)+\left(\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{125}\right)+\left(-\dfrac{19}{5}\right)\sqrt{45}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(0+\left(-\dfrac{3}{2}\right)\sqrt{125}+3+\dfrac{41}{4}+\dfrac{6}{5}+\left(\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{77}{6}\right)\sqrt{25}+\dfrac{41}{4}\right)+\left(\left(\left(2\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{45}\right)+\left(\dfrac{23}{7}\right)\sqrt{125}-\dfrac{51}{5}-\left(\left(\dfrac{29}{2}\right)\sqrt{45}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{59}{2}\right)\sqrt{125}\right)+\left(\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{125}\right)+\left(-\dfrac{19}{5}\right)\sqrt{45}\right)\\
&=&\left(0+\left(-\dfrac{15}{2}\right)\sqrt{5}+3+\dfrac{41}{4}+\dfrac{6}{5}+\dfrac{385}{6}+\dfrac{385}{6}+\dfrac{41}{4}\right)+\left(10-\left(\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)+\left(\dfrac{115}{7}\right)\sqrt{5}-\dfrac{51}{5}-\left(\left(\dfrac{87}{2}\right)\sqrt{5}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{295}{2}\right)\sqrt{5}\right)+\left(\left(30\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{5}\right)+\left(-\dfrac{57}{5}\right)\sqrt{5}\right)\\
&=&0+\left(-\dfrac{15}{2}\right)\sqrt{5}+3+\dfrac{41}{4}+\dfrac{6}{5}+\dfrac{385}{6}+\dfrac{385}{6}+\dfrac{41}{4}+10-\left(\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)+\left(\dfrac{115}{7}\right)\sqrt{5}-\dfrac{51}{5}-\left(\left(\dfrac{87}{2}\right)\sqrt{5}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{295}{2}\right)\sqrt{5}\right)+\left(\left(30\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{5}\right)+\left(-\dfrac{57}{5}\right)\sqrt{5}\\
&=&\dfrac{2761}{18}+\left(-\dfrac{6999}{35}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(0+\left(-\dfrac{3}{2}\right)\sqrt{125}+3+\dfrac{41}{4}+\dfrac{6}{5}+\left(\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{77}{6}\right)\sqrt{25}+\dfrac{41}{4}\right)-\left(\left(\left(2\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{45}\right)+\left(\dfrac{23}{7}\right)\sqrt{125}-\dfrac{51}{5}-\left(\left(\dfrac{29}{2}\right)\sqrt{45}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{59}{2}\right)\sqrt{125}\right)+\left(\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{125}\right)+\left(-\dfrac{19}{5}\right)\sqrt{45}\right)\\
&=&\left(0+\left(-\dfrac{15}{2}\right)\sqrt{5}+3+\dfrac{41}{4}+\dfrac{6}{5}+\dfrac{385}{6}+\dfrac{385}{6}+\dfrac{41}{4}\right)-\left(10-\left(\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)+\left(\dfrac{115}{7}\right)\sqrt{5}-\dfrac{51}{5}-\left(\left(\dfrac{87}{2}\right)\sqrt{5}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{295}{2}\right)\sqrt{5}\right)+\left(\left(30\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{5}\right)+\left(-\dfrac{57}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{4591}{30}+\left(-\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\dfrac{16}{45}+\left(-\dfrac{13473}{70}\right)\sqrt{5}\right)\\
&=&\dfrac{4591}{30}+\left(-\dfrac{15}{2}\right)\sqrt{5}+-\dfrac{16}{45}+\left(\dfrac{13473}{70}\right)\sqrt{5}\\
&=&\dfrac{13741}{90}+\left(\dfrac{6474}{35}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(0+\left(-\dfrac{3}{2}\right)\sqrt{125}+3+\dfrac{41}{4}+\dfrac{6}{5}+\left(\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{77}{6}\right)\sqrt{25}+\dfrac{41}{4}\right)\times\left(\left(\left(2\right)\sqrt{25}\right)-\left(\left(9\right)\sqrt{45}\right)-\left(\left(\dfrac{23}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{45}\right)+\left(\dfrac{23}{7}\right)\sqrt{125}-\dfrac{51}{5}-\left(\left(\dfrac{29}{2}\right)\sqrt{45}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{59}{2}\right)\sqrt{125}\right)+\left(\left(6\right)\sqrt{125}\right)-\left(\left(-\dfrac{7}{2}\right)\sqrt{125}\right)+\left(-\dfrac{19}{5}\right)\sqrt{45}\right)\\
&=&\left(0+\left(-\dfrac{15}{2}\right)\sqrt{5}+3+\dfrac{41}{4}+\dfrac{6}{5}+\dfrac{385}{6}+\dfrac{385}{6}+\dfrac{41}{4}\right)\times\left(10-\left(\left(27\right)\sqrt{5}\right)-\left(\left(\dfrac{46}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{35}{3}\right)\sqrt{5}\right)+\left(\dfrac{115}{7}\right)\sqrt{5}-\dfrac{51}{5}-\left(\left(\dfrac{87}{2}\right)\sqrt{5}\right)-\dfrac{61}{9}+\dfrac{22}{3}-\left(\left(\dfrac{295}{2}\right)\sqrt{5}\right)+\left(\left(30\right)\sqrt{5}\right)-\left(\left(-\dfrac{35}{2}\right)\sqrt{5}\right)+\left(-\dfrac{57}{5}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{4591}{30}+\left(-\dfrac{15}{2}\right)\sqrt{5}\right)\left(\dfrac{16}{45}+\left(-\dfrac{13473}{70}\right)\sqrt{5}\right)\\
&=&\dfrac{36728}{675}+\left(-\dfrac{61860143}{2100}\right)\sqrt{5}+\left(\dfrac{40419}{28}\right)\sqrt{25}\\
&=&\dfrac{137442509}{18900}+\left(-\dfrac{61860143}{2100}\right)\sqrt{5}\\
\end{eqnarray*}