L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{65}{4}\right)\sqrt{45}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(9\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{20}\right)+8\right)\) et \( Y=\dfrac{18}{5}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{65}{4}\right)\sqrt{45}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(9\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{20}\right)+8\right)\right)+\left(\dfrac{18}{5}\right)\\
&=&\left(\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(27\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{2}\right)\sqrt{5}\right)\right)-\left(\left(\left(-25\right)\sqrt{5}\right)+8\right)\right)+\left(\dfrac{18}{5}\right)\\
&=&\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(27\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{2}\right)\sqrt{5}\right)\right)-\left(\left(\left(-25\right)\sqrt{5}\right)+8\right)+\dfrac{18}{5}\\
&=&-\dfrac{89}{60}+\left(\dfrac{101}{4}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{65}{4}\right)\sqrt{45}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(9\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{20}\right)+8\right)\right)-\left(\dfrac{18}{5}\right)\\
&=&\left(\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(27\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{2}\right)\sqrt{5}\right)\right)-\left(\left(\left(-25\right)\sqrt{5}\right)+8\right)\right)-\left(\dfrac{18}{5}\right)\\
&=&\left(-\dfrac{61}{12}+\left(\dfrac{101}{4}\right)\sqrt{5}\right)-\left(\dfrac{18}{5}\right)\\
&=&-\dfrac{61}{12}+\left(\dfrac{101}{4}\right)\sqrt{5}+-\dfrac{18}{5}\\
&=&-\dfrac{521}{60}+\left(\dfrac{101}{4}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{65}{4}\right)\sqrt{45}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(9\right)\sqrt{45}\right)-\left(\left(-4\right)\sqrt{125}\right)-\left(\left(\dfrac{37}{4}\right)\sqrt{20}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{25}{2}\right)\sqrt{20}\right)+8\right)\right)\times\left(\dfrac{18}{5}\right)\\
&=&\left(\dfrac{5}{8}-\left(-9-\left(\left(-\dfrac{195}{4}\right)\sqrt{5}\right)-\dfrac{5}{8}+\dfrac{22}{3}\right)-\left(\left(\left(27\right)\sqrt{5}\right)-\left(\left(-20\right)\sqrt{5}\right)-\left(\left(\dfrac{37}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{155}{2}\right)\sqrt{5}\right)\right)-\left(\left(\left(-25\right)\sqrt{5}\right)+8\right)\right)\times\left(\dfrac{18}{5}\right)\\
&=&\left(-\dfrac{61}{12}+\left(\dfrac{101}{4}\right)\sqrt{5}\right)\left(\dfrac{18}{5}\right)\\
&=&-\dfrac{183}{10}+\left(\dfrac{909}{10}\right)\sqrt{5}\\
&=&-\dfrac{183}{10}+\left(\dfrac{909}{10}\right)\sqrt{5}\\
\end{eqnarray*}