L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{77}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{20}\right)\right)-\left(-8-\left(\left(-1\right)\sqrt{25}\right)\right)-\left(\left(\left(\dfrac{17}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{45}\right)\right)-\left(-9+\left(\dfrac{11}{3}\right)\sqrt{25}+\dfrac{1}{7}\right)\) et \( Y=-\dfrac{5}{6}-5+\left(\dfrac{1}{3}\right)\sqrt{45}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(\dfrac{77}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{20}\right)\right)-\left(-8-\left(\left(-1\right)\sqrt{25}\right)\right)-\left(\left(\left(\dfrac{17}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{45}\right)\right)-\left(-9+\left(\dfrac{11}{3}\right)\sqrt{25}+\dfrac{1}{7}\right)\right)+\left(-\dfrac{5}{6}-5+\left(\dfrac{1}{3}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\left(\dfrac{231}{8}\right)\sqrt{5}\right)-\left(\left(65\right)\sqrt{5}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{5}\right)\right)-\left(-8+5\right)-\left(\left(\left(\dfrac{17}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{5}\right)\right)-\left(-9+\dfrac{55}{3}+\dfrac{1}{7}\right)\right)+\left(-\dfrac{5}{6}-5+\left(1\right)\sqrt{5}\right)\\
&=&\left(\left(\left(\dfrac{231}{8}\right)\sqrt{5}\right)-\left(\left(65\right)\sqrt{5}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{5}\right)\right)-\left(-8+5\right)-\left(\left(\left(\dfrac{17}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{5}\right)\right)-\left(-9+\dfrac{55}{3}+\dfrac{1}{7}\right)-\dfrac{5}{6}-5+\left(1\right)\sqrt{5}\\
&=&\left(-\dfrac{781}{24}\right)\sqrt{5}-\dfrac{517}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{77}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{20}\right)\right)-\left(-8-\left(\left(-1\right)\sqrt{25}\right)\right)-\left(\left(\left(\dfrac{17}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{45}\right)\right)-\left(-9+\left(\dfrac{11}{3}\right)\sqrt{25}+\dfrac{1}{7}\right)\right)-\left(-\dfrac{5}{6}-5+\left(\dfrac{1}{3}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\left(\dfrac{231}{8}\right)\sqrt{5}\right)-\left(\left(65\right)\sqrt{5}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{5}\right)\right)-\left(-8+5\right)-\left(\left(\left(\dfrac{17}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{5}\right)\right)-\left(-9+\dfrac{55}{3}+\dfrac{1}{7}\right)\right)-\left(-\dfrac{5}{6}-5+\left(1\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{805}{24}\right)\sqrt{5}-\dfrac{136}{21}\right)-\left(-\dfrac{35}{6}+\left(1\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{805}{24}\right)\sqrt{5}-\dfrac{136}{21}+\dfrac{35}{6}+\left(-1\right)\sqrt{5}\\
&=&\left(-\dfrac{829}{24}\right)\sqrt{5}-\dfrac{9}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{77}{8}\right)\sqrt{45}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{20}\right)\right)-\left(-8-\left(\left(-1\right)\sqrt{25}\right)\right)-\left(\left(\left(\dfrac{17}{9}\right)\sqrt{45}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{45}\right)\right)-\left(-9+\left(\dfrac{11}{3}\right)\sqrt{25}+\dfrac{1}{7}\right)\right)\times\left(-\dfrac{5}{6}-5+\left(\dfrac{1}{3}\right)\sqrt{45}\right)\\
&=&\left(\left(\left(\left(\dfrac{231}{8}\right)\sqrt{5}\right)-\left(\left(65\right)\sqrt{5}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{5}\right)\right)-\left(-8+5\right)-\left(\left(\left(\dfrac{17}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{29}{2}\right)\sqrt{5}\right)\right)-\left(-9+\dfrac{55}{3}+\dfrac{1}{7}\right)\right)\times\left(-\dfrac{5}{6}-5+\left(1\right)\sqrt{5}\right)\\
&=&\left(\left(-\dfrac{805}{24}\right)\sqrt{5}-\dfrac{136}{21}\right)\left(-\dfrac{35}{6}+\left(1\right)\sqrt{5}\right)\\
&=&\left(\dfrac{190697}{1008}\right)\sqrt{5}+\left(-\dfrac{805}{24}\right)\sqrt{25}+\dfrac{340}{9}\\
&=&\left(\dfrac{190697}{1008}\right)\sqrt{5}-\dfrac{9355}{72}\\
\end{eqnarray*}