L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{1}{2}\right)\sqrt{18}\right)+\dfrac{51}{7}\right)-\left(\left(3\right)\sqrt{4}+\left(-\dfrac{9}{2}\right)\sqrt{4}\right)+\dfrac{33}{4}-\left(\left(\dfrac{40}{3}\right)\sqrt{50}\right)\) et \( Y=\left(\left(\dfrac{11}{3}\right)\sqrt{4}+\dfrac{1}{2}+\left(-8\right)\sqrt{18}+\left(-\dfrac{5}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{50}\right)\) . 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{1}{2}\right)\sqrt{18}\right)+\dfrac{51}{7}\right)-\left(\left(3\right)\sqrt{4}+\left(-\dfrac{9}{2}\right)\sqrt{4}\right)+\dfrac{33}{4}-\left(\left(\dfrac{40}{3}\right)\sqrt{50}\right)\right)+\left(\left(\left(\dfrac{11}{3}\right)\sqrt{4}+\dfrac{1}{2}+\left(-8\right)\sqrt{18}+\left(-\dfrac{5}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{3}{2}\right)\sqrt{2}\right)+\dfrac{51}{7}\right)-\left(6-9\right)+\dfrac{33}{4}-\left(\left(\dfrac{200}{3}\right)\sqrt{2}\right)\right)+\left(\left(\dfrac{22}{3}+\dfrac{1}{2}+\left(-24\right)\sqrt{2}-\dfrac{5}{2}\right)-\dfrac{75}{4}-\left(\left(\dfrac{115}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\left(\dfrac{3}{2}\right)\sqrt{2}\right)+\dfrac{51}{7}\right)-\left(6-9\right)+\dfrac{33}{4}-\left(\left(\dfrac{200}{3}\right)\sqrt{2}\right)+\left(\dfrac{22}{3}+\dfrac{1}{2}+\left(-24\right)\sqrt{2}-\dfrac{5}{2}\right)-\dfrac{75}{4}-\left(\left(\dfrac{115}{9}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{1835}{18}\right)\sqrt{2}+\dfrac{215}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{1}{2}\right)\sqrt{18}\right)+\dfrac{51}{7}\right)-\left(\left(3\right)\sqrt{4}+\left(-\dfrac{9}{2}\right)\sqrt{4}\right)+\dfrac{33}{4}-\left(\left(\dfrac{40}{3}\right)\sqrt{50}\right)\right)-\left(\left(\left(\dfrac{11}{3}\right)\sqrt{4}+\dfrac{1}{2}+\left(-8\right)\sqrt{18}+\left(-\dfrac{5}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{3}{2}\right)\sqrt{2}\right)+\dfrac{51}{7}\right)-\left(6-9\right)+\dfrac{33}{4}-\left(\left(\dfrac{200}{3}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{22}{3}+\dfrac{1}{2}+\left(-24\right)\sqrt{2}-\dfrac{5}{2}\right)-\dfrac{75}{4}-\left(\left(\dfrac{115}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{391}{6}\right)\sqrt{2}+\dfrac{519}{28}\right)-\left(-\dfrac{161}{12}+\left(-\dfrac{331}{9}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{391}{6}\right)\sqrt{2}+\dfrac{519}{28}+\dfrac{161}{12}+\left(\dfrac{331}{9}\right)\sqrt{2}\\
&=&\left(-\dfrac{511}{18}\right)\sqrt{2}+\dfrac{671}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{1}{2}\right)\sqrt{18}\right)+\dfrac{51}{7}\right)-\left(\left(3\right)\sqrt{4}+\left(-\dfrac{9}{2}\right)\sqrt{4}\right)+\dfrac{33}{4}-\left(\left(\dfrac{40}{3}\right)\sqrt{50}\right)\right)\times\left(\left(\left(\dfrac{11}{3}\right)\sqrt{4}+\dfrac{1}{2}+\left(-8\right)\sqrt{18}+\left(-\dfrac{5}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{75}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{23}{9}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{3}{2}\right)\sqrt{2}\right)+\dfrac{51}{7}\right)-\left(6-9\right)+\dfrac{33}{4}-\left(\left(\dfrac{200}{3}\right)\sqrt{2}\right)\right)\times\left(\left(\dfrac{22}{3}+\dfrac{1}{2}+\left(-24\right)\sqrt{2}-\dfrac{5}{2}\right)-\dfrac{75}{4}-\left(\left(\dfrac{115}{9}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{391}{6}\right)\sqrt{2}+\dfrac{519}{28}\right)\left(-\dfrac{161}{12}+\left(-\dfrac{331}{9}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{97079}{504}\right)\sqrt{2}+\left(\dfrac{129421}{54}\right)\sqrt{4}-\dfrac{3979}{16}\\
&=&\left(\dfrac{97079}{504}\right)\sqrt{2}+\dfrac{1963303}{432}\\
\end{eqnarray*}