L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{24}{7}\) et \( Y=\left(\left(8\right)\sqrt{50}\right)-\dfrac{31}{4}-\left(\left(3\right)\sqrt{4}\right)+\left(-\dfrac{71}{2}\right)\sqrt{50}+\left(-5\right)\sqrt{18}-\dfrac{13}{4}-\dfrac{14}{3}+\left(-\dfrac{31}{3}\right)\sqrt{4}+\left(-\dfrac{5}{2}\right)\sqrt{18}+\left(\dfrac{47}{5}\right)\sqrt{18}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-\dfrac{23}{2}\right)\sqrt{8}+\left(-4\right)\sqrt{18}-9+\dfrac{1}{4}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(\dfrac{79}{5}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{24}{7}\right)+\left(\left(\left(8\right)\sqrt{50}\right)-\dfrac{31}{4}-\left(\left(3\right)\sqrt{4}\right)+\left(-\dfrac{71}{2}\right)\sqrt{50}+\left(-5\right)\sqrt{18}-\dfrac{13}{4}-\dfrac{14}{3}+\left(-\dfrac{31}{3}\right)\sqrt{4}+\left(-\dfrac{5}{2}\right)\sqrt{18}+\left(\dfrac{47}{5}\right)\sqrt{18}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-\dfrac{23}{2}\right)\sqrt{8}+\left(-4\right)\sqrt{18}-9+\dfrac{1}{4}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(\dfrac{79}{5}\right)\sqrt{50}\right)\\
&=&\left(-\dfrac{24}{7}\right)+\left(\left(\left(40\right)\sqrt{2}\right)-\dfrac{31}{4}-6+\left(-\dfrac{355}{2}\right)\sqrt{2}+\left(-15\right)\sqrt{2}-\dfrac{13}{4}-\dfrac{14}{3}-\dfrac{62}{3}+\left(-\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{141}{5}\right)\sqrt{2}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-23\right)\sqrt{2}+\left(-12\right)\sqrt{2}-9+\dfrac{1}{4}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(79\right)\sqrt{2}\right)\\
&=&-\dfrac{24}{7}+\left(\left(40\right)\sqrt{2}\right)-\dfrac{31}{4}-6+\left(-\dfrac{355}{2}\right)\sqrt{2}+\left(-15\right)\sqrt{2}-\dfrac{13}{4}-\dfrac{14}{3}-\dfrac{62}{3}+\left(-\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{141}{5}\right)\sqrt{2}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-23\right)\sqrt{2}+\left(-12\right)\sqrt{2}-9+\dfrac{1}{4}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(79\right)\sqrt{2}\\
&=&-\dfrac{8801}{168}+\left(-\dfrac{2981}{20}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{24}{7}\right)-\left(\left(\left(8\right)\sqrt{50}\right)-\dfrac{31}{4}-\left(\left(3\right)\sqrt{4}\right)+\left(-\dfrac{71}{2}\right)\sqrt{50}+\left(-5\right)\sqrt{18}-\dfrac{13}{4}-\dfrac{14}{3}+\left(-\dfrac{31}{3}\right)\sqrt{4}+\left(-\dfrac{5}{2}\right)\sqrt{18}+\left(\dfrac{47}{5}\right)\sqrt{18}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-\dfrac{23}{2}\right)\sqrt{8}+\left(-4\right)\sqrt{18}-9+\dfrac{1}{4}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(\dfrac{79}{5}\right)\sqrt{50}\right)\\
&=&\left(-\dfrac{24}{7}\right)-\left(\left(\left(40\right)\sqrt{2}\right)-\dfrac{31}{4}-6+\left(-\dfrac{355}{2}\right)\sqrt{2}+\left(-15\right)\sqrt{2}-\dfrac{13}{4}-\dfrac{14}{3}-\dfrac{62}{3}+\left(-\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{141}{5}\right)\sqrt{2}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-23\right)\sqrt{2}+\left(-12\right)\sqrt{2}-9+\dfrac{1}{4}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(79\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{24}{7}\right)-\left(\left(-\dfrac{2981}{20}\right)\sqrt{2}-\dfrac{1175}{24}\right)\\
&=&-\dfrac{24}{7}+\left(\dfrac{2981}{20}\right)\sqrt{2}+\dfrac{1175}{24}\\
&=&\dfrac{7649}{168}+\left(\dfrac{2981}{20}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{24}{7}\right)\times\left(\left(\left(8\right)\sqrt{50}\right)-\dfrac{31}{4}-\left(\left(3\right)\sqrt{4}\right)+\left(-\dfrac{71}{2}\right)\sqrt{50}+\left(-5\right)\sqrt{18}-\dfrac{13}{4}-\dfrac{14}{3}+\left(-\dfrac{31}{3}\right)\sqrt{4}+\left(-\dfrac{5}{2}\right)\sqrt{18}+\left(\dfrac{47}{5}\right)\sqrt{18}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-\dfrac{23}{2}\right)\sqrt{8}+\left(-4\right)\sqrt{18}-9+\dfrac{1}{4}+\left(-\dfrac{49}{4}\right)\sqrt{50}+\left(\dfrac{79}{5}\right)\sqrt{50}\right)\\
&=&\left(-\dfrac{24}{7}\right)\times\left(\left(\left(40\right)\sqrt{2}\right)-\dfrac{31}{4}-6+\left(-\dfrac{355}{2}\right)\sqrt{2}+\left(-15\right)\sqrt{2}-\dfrac{13}{4}-\dfrac{14}{3}-\dfrac{62}{3}+\left(-\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{141}{5}\right)\sqrt{2}-\dfrac{59}{8}+\dfrac{19}{2}+\left(-23\right)\sqrt{2}+\left(-12\right)\sqrt{2}-9+\dfrac{1}{4}+\left(-\dfrac{245}{4}\right)\sqrt{2}+\left(79\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{24}{7}\right)\left(\left(-\dfrac{2981}{20}\right)\sqrt{2}-\dfrac{1175}{24}\right)\\
&=&\left(\dfrac{17886}{35}\right)\sqrt{2}+\dfrac{1175}{7}\\
&=&\left(\dfrac{17886}{35}\right)\sqrt{2}+\dfrac{1175}{7}\\
\end{eqnarray*}