L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{45}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{20}\right)\) et \( Y=\left(\left(-\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{45}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{20}\right)\right)+\left(\left(\left(-\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{5}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{4}\right)\sqrt{5}\right)\right)+\left(\left(\left(-\dfrac{35}{3}\right)\sqrt{5}\right)-\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{8}\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{115}{4}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{5}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{4}\right)\sqrt{5}\right)+\left(\left(-\dfrac{35}{3}\right)\sqrt{5}\right)-\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{8}\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{115}{4}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{527}{24}\right)\sqrt{5}+\dfrac{62}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{45}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{20}\right)\right)-\left(\left(\left(-\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{5}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{4}\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{35}{3}\right)\sqrt{5}\right)-\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{8}\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{115}{4}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(-\dfrac{49}{4}\right)\sqrt{5}+\dfrac{62}{9}\right)-\left(\left(\dfrac{821}{24}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{49}{4}\right)\sqrt{5}+\dfrac{62}{9}+\left(-\dfrac{821}{24}\right)\sqrt{5}\\
&=&\left(-\dfrac{1115}{24}\right)\sqrt{5}+\dfrac{62}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{19}{2}\right)\sqrt{45}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{20}\right)\right)\times\left(\left(\left(-\dfrac{7}{3}\right)\sqrt{125}\right)-\left(\left(\left(9\right)\sqrt{125}\right)-\left(\left(\dfrac{61}{8}\right)\sqrt{125}\right)\right)-\left(\left(\left(-\dfrac{23}{4}\right)\sqrt{125}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{57}{2}\right)\sqrt{5}\right)+\dfrac{62}{9}\right)-\left(\left(-\dfrac{65}{4}\right)\sqrt{5}\right)\right)\times\left(\left(\left(-\dfrac{35}{3}\right)\sqrt{5}\right)-\left(\left(\left(45\right)\sqrt{5}\right)-\left(\left(\dfrac{305}{8}\right)\sqrt{5}\right)\right)-\left(\left(\left(-\dfrac{115}{4}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(-\dfrac{49}{4}\right)\sqrt{5}+\dfrac{62}{9}\right)\left(\left(\dfrac{821}{24}\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{40229}{96}\right)\sqrt{25}+\left(\dfrac{25451}{108}\right)\sqrt{5}\\
&=&-\dfrac{201145}{96}+\left(\dfrac{25451}{108}\right)\sqrt{5}\\
\end{eqnarray*}