L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{34}{5}\right)\sqrt{25}\right)-\dfrac{33}{4}\right)-\left(\left(\left(-5\right)\sqrt{25}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{45}\right)\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{25}\right)\right)\) et \( Y=\left(\dfrac{51}{5}\right)\sqrt{125}+\left(-\dfrac{29}{9}\right)\sqrt{125}-\dfrac{17}{2}-\left(\left(\dfrac{61}{2}\right)\sqrt{45}\right)\) . 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{34}{5}\right)\sqrt{25}\right)-\dfrac{33}{4}\right)-\left(\left(\left(-5\right)\sqrt{25}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{45}\right)\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{25}\right)\right)\right)+\left(\left(\dfrac{51}{5}\right)\sqrt{125}+\left(-\dfrac{29}{9}\right)\sqrt{125}-\dfrac{17}{2}-\left(\left(\dfrac{61}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(-34-\dfrac{33}{4}\right)-\left(-25-\left(\left(-\dfrac{165}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{385}{3}-\dfrac{25}{7}\right)\right)+\left(\left(51\right)\sqrt{5}+\left(-\dfrac{145}{9}\right)\sqrt{5}-\dfrac{17}{2}-\left(\left(\dfrac{183}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(-34-\dfrac{33}{4}\right)-\left(-25-\left(\left(-\dfrac{165}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{385}{3}-\dfrac{25}{7}\right)+\left(51\right)\sqrt{5}+\left(-\dfrac{145}{9}\right)\sqrt{5}-\dfrac{17}{2}-\left(\left(\dfrac{183}{2}\right)\sqrt{5}\right)\\
&=&-\dfrac{12643}{84}+\left(-\dfrac{10103}{126}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{34}{5}\right)\sqrt{25}\right)-\dfrac{33}{4}\right)-\left(\left(\left(-5\right)\sqrt{25}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{45}\right)\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{25}\right)\right)\right)-\left(\left(\dfrac{51}{5}\right)\sqrt{125}+\left(-\dfrac{29}{9}\right)\sqrt{125}-\dfrac{17}{2}-\left(\left(\dfrac{61}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(-34-\dfrac{33}{4}\right)-\left(-25-\left(\left(-\dfrac{165}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{385}{3}-\dfrac{25}{7}\right)\right)-\left(\left(51\right)\sqrt{5}+\left(-\dfrac{145}{9}\right)\sqrt{5}-\dfrac{17}{2}-\left(\left(\dfrac{183}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{11929}{84}+\left(-\dfrac{165}{7}\right)\sqrt{5}\right)-\left(\left(-\dfrac{1019}{18}\right)\sqrt{5}-\dfrac{17}{2}\right)\\
&=&-\dfrac{11929}{84}+\left(-\dfrac{165}{7}\right)\sqrt{5}+\left(\dfrac{1019}{18}\right)\sqrt{5}+\dfrac{17}{2}\\
&=&-\dfrac{11215}{84}+\left(\dfrac{4163}{126}\right)\sqrt{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{34}{5}\right)\sqrt{25}\right)-\dfrac{33}{4}\right)-\left(\left(\left(-5\right)\sqrt{25}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{45}\right)\right)-\left(\left(\left(\dfrac{77}{3}\right)\sqrt{25}\right)-\left(\left(\dfrac{5}{7}\right)\sqrt{25}\right)\right)\right)\times\left(\left(\dfrac{51}{5}\right)\sqrt{125}+\left(-\dfrac{29}{9}\right)\sqrt{125}-\dfrac{17}{2}-\left(\left(\dfrac{61}{2}\right)\sqrt{45}\right)\right)\\
&=&\left(\left(-34-\dfrac{33}{4}\right)-\left(-25-\left(\left(-\dfrac{165}{7}\right)\sqrt{5}\right)\right)-\left(\dfrac{385}{3}-\dfrac{25}{7}\right)\right)\times\left(\left(51\right)\sqrt{5}+\left(-\dfrac{145}{9}\right)\sqrt{5}-\dfrac{17}{2}-\left(\left(\dfrac{183}{2}\right)\sqrt{5}\right)\right)\\
&=&\left(-\dfrac{11929}{84}+\left(-\dfrac{165}{7}\right)\sqrt{5}\right)\left(\left(-\dfrac{1019}{18}\right)\sqrt{5}-\dfrac{17}{2}\right)\\
&=&\left(\dfrac{12458591}{1512}\right)\sqrt{5}+\dfrac{202793}{168}+\left(\dfrac{56045}{42}\right)\sqrt{25}\\
&=&\left(\dfrac{12458591}{1512}\right)\sqrt{5}+\dfrac{63033}{8}\\
\end{eqnarray*}