L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{53}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{7}\right)\sqrt{45}\right)-\left(\left(-5\right)\sqrt{20}\right)-\left(\left(7\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}+\dfrac{23}{6}+\left(\dfrac{59}{4}\right)\sqrt{125}-\dfrac{49}{4}\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{25}\right)+1-\left(\left(9\right)\sqrt{25}\right)-\dfrac{23}{7}-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)\) et \( Y=\left(-\dfrac{25}{3}\right)\sqrt{125}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(\dfrac{53}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{7}\right)\sqrt{45}\right)-\left(\left(-5\right)\sqrt{20}\right)-\left(\left(7\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}+\dfrac{23}{6}+\left(\dfrac{59}{4}\right)\sqrt{125}-\dfrac{49}{4}\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{25}\right)+1-\left(\left(9\right)\sqrt{25}\right)-\dfrac{23}{7}-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)\right)+\left(\left(-\dfrac{25}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(\left(\left(\dfrac{159}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{7}\right)\sqrt{5}\right)-\left(\left(-10\right)\sqrt{5}\right)-35-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}+\dfrac{23}{6}+\left(\dfrac{295}{4}\right)\sqrt{5}-\dfrac{49}{4}\right)-\left(-41+1-45-\dfrac{23}{7}-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)\right)+\left(\left(-\dfrac{125}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\left(\dfrac{159}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{7}\right)\sqrt{5}\right)-\left(\left(-10\right)\sqrt{5}\right)-35-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}+\dfrac{23}{6}+\left(\dfrac{295}{4}\right)\sqrt{5}-\dfrac{49}{4}\right)-\left(-41+1-45-\dfrac{23}{7}-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)+\left(-\dfrac{125}{3}\right)\sqrt{5}\\
&=&\left(\dfrac{4003}{84}\right)\sqrt{5}+\dfrac{5183}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{53}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{7}\right)\sqrt{45}\right)-\left(\left(-5\right)\sqrt{20}\right)-\left(\left(7\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}+\dfrac{23}{6}+\left(\dfrac{59}{4}\right)\sqrt{125}-\dfrac{49}{4}\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{25}\right)+1-\left(\left(9\right)\sqrt{25}\right)-\dfrac{23}{7}-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(\left(\left(\dfrac{159}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{7}\right)\sqrt{5}\right)-\left(\left(-10\right)\sqrt{5}\right)-35-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}+\dfrac{23}{6}+\left(\dfrac{295}{4}\right)\sqrt{5}-\dfrac{49}{4}\right)-\left(-41+1-45-\dfrac{23}{7}-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)\right)-\left(\left(-\dfrac{125}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{2501}{28}\right)\sqrt{5}+\dfrac{5183}{84}\right)-\left(\left(-\dfrac{125}{3}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{2501}{28}\right)\sqrt{5}+\dfrac{5183}{84}+\left(\dfrac{125}{3}\right)\sqrt{5}\\
&=&\left(\dfrac{11003}{84}\right)\sqrt{5}+\dfrac{5183}{84}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{53}{7}\right)\sqrt{45}\right)-\left(\left(-\dfrac{37}{7}\right)\sqrt{45}\right)-\left(\left(-5\right)\sqrt{20}\right)-\left(\left(7\right)\sqrt{25}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{20}\right)\right)-\left(\left(-\dfrac{77}{8}\right)\sqrt{125}+\dfrac{23}{6}+\left(\dfrac{59}{4}\right)\sqrt{125}-\dfrac{49}{4}\right)-\left(\left(\left(-\dfrac{41}{5}\right)\sqrt{25}\right)+1-\left(\left(9\right)\sqrt{25}\right)-\dfrac{23}{7}-\left(\left(\dfrac{63}{8}\right)\sqrt{125}\right)\right)\right)\times\left(\left(-\dfrac{25}{3}\right)\sqrt{125}\right)\\
&=&\left(\left(\left(\left(\dfrac{159}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{111}{7}\right)\sqrt{5}\right)-\left(\left(-10\right)\sqrt{5}\right)-35-\left(\left(-27\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{385}{8}\right)\sqrt{5}+\dfrac{23}{6}+\left(\dfrac{295}{4}\right)\sqrt{5}-\dfrac{49}{4}\right)-\left(-41+1-45-\dfrac{23}{7}-\left(\left(\dfrac{315}{8}\right)\sqrt{5}\right)\right)\right)\times\left(\left(-\dfrac{125}{3}\right)\sqrt{5}\right)\\
&=&\left(\left(\dfrac{2501}{28}\right)\sqrt{5}+\dfrac{5183}{84}\right)\left(\left(-\dfrac{125}{3}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{312625}{84}\right)\sqrt{25}+\left(-\dfrac{647875}{252}\right)\sqrt{5}\\
&=&-\dfrac{1563125}{84}+\left(-\dfrac{647875}{252}\right)\sqrt{5}\\
\end{eqnarray*}