L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{29}{3}\right)\sqrt{18}+\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)+\left(-1\right)\sqrt{4}+\left(-\dfrac{29}{3}\right)\sqrt{18}+\dfrac{53}{6}\) et \( Y=\left(-2+\left(\dfrac{11}{7}\right)\sqrt{50}+\left(\dfrac{59}{7}\right)\sqrt{18}+\left(\dfrac{5}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)-\left(-\dfrac{11}{3}+\left(-4\right)\sqrt{8}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{29}{3}\right)\sqrt{18}+\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)+\left(-1\right)\sqrt{4}+\left(-\dfrac{29}{3}\right)\sqrt{18}+\dfrac{53}{6}\right)+\left(\left(-2+\left(\dfrac{11}{7}\right)\sqrt{50}+\left(\dfrac{59}{7}\right)\sqrt{18}+\left(\dfrac{5}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)-\left(-\dfrac{11}{3}+\left(-4\right)\sqrt{8}\right)\right)\\
&=&\left(\left(-29\right)\sqrt{2}+\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{98}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+27-2+\left(-29\right)\sqrt{2}+\dfrac{53}{6}\right)+\left(\left(-2+\left(\dfrac{55}{7}\right)\sqrt{2}+\left(\dfrac{177}{7}\right)\sqrt{2}+\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{230}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)-\left(-\dfrac{11}{3}+\left(-8\right)\sqrt{2}\right)\right)\\
&=&\left(-29\right)\sqrt{2}+\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{98}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+27-2+\left(-29\right)\sqrt{2}+\dfrac{53}{6}+\left(-2+\left(\dfrac{55}{7}\right)\sqrt{2}+\left(\dfrac{177}{7}\right)\sqrt{2}+\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{230}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)-\left(-\dfrac{11}{3}+\left(-8\right)\sqrt{2}\right)\\
&=&\left(\dfrac{3343}{63}\right)\sqrt{2}+\dfrac{95}{6}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{29}{3}\right)\sqrt{18}+\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)+\left(-1\right)\sqrt{4}+\left(-\dfrac{29}{3}\right)\sqrt{18}+\dfrac{53}{6}\right)-\left(\left(-2+\left(\dfrac{11}{7}\right)\sqrt{50}+\left(\dfrac{59}{7}\right)\sqrt{18}+\left(\dfrac{5}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)-\left(-\dfrac{11}{3}+\left(-4\right)\sqrt{8}\right)\right)\\
&=&\left(\left(-29\right)\sqrt{2}+\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{98}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+27-2+\left(-29\right)\sqrt{2}+\dfrac{53}{6}\right)-\left(\left(-2+\left(\dfrac{55}{7}\right)\sqrt{2}+\left(\dfrac{177}{7}\right)\sqrt{2}+\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{230}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)-\left(-\dfrac{11}{3}+\left(-8\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{2}+\dfrac{85}{6}\right)-\left(\dfrac{5}{3}+\left(\dfrac{429}{7}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{74}{9}\right)\sqrt{2}+\dfrac{85}{6}+-\dfrac{5}{3}+\left(-\dfrac{429}{7}\right)\sqrt{2}\\
&=&\left(-\dfrac{4379}{63}\right)\sqrt{2}+\dfrac{25}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{29}{3}\right)\sqrt{18}+\left(\left(\dfrac{77}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{49}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{59}{6}\right)\sqrt{4}\right)-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)+\left(-1\right)\sqrt{4}+\left(-\dfrac{29}{3}\right)\sqrt{18}+\dfrac{53}{6}\right)\times\left(\left(-2+\left(\dfrac{11}{7}\right)\sqrt{50}+\left(\dfrac{59}{7}\right)\sqrt{18}+\left(\dfrac{5}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{46}{7}\right)\sqrt{50}\right)-\left(\left(\dfrac{62}{7}\right)\sqrt{8}\right)-\left(-\dfrac{11}{3}+\left(-4\right)\sqrt{8}\right)\right)\\
&=&\left(\left(-29\right)\sqrt{2}+\left(\left(\dfrac{154}{9}\right)\sqrt{2}\right)-\left(\left(-\dfrac{98}{3}\right)\sqrt{2}\right)-\dfrac{59}{3}+27-2+\left(-29\right)\sqrt{2}+\dfrac{53}{6}\right)\times\left(\left(-2+\left(\dfrac{55}{7}\right)\sqrt{2}+\left(\dfrac{177}{7}\right)\sqrt{2}+\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{230}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{124}{7}\right)\sqrt{2}\right)-\left(-\dfrac{11}{3}+\left(-8\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{74}{9}\right)\sqrt{2}+\dfrac{85}{6}\right)\left(\dfrac{5}{3}+\left(\dfrac{429}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{323005}{378}\right)\sqrt{2}+\left(-\dfrac{10582}{21}\right)\sqrt{4}+\dfrac{425}{18}\\
&=&\left(\dfrac{323005}{378}\right)\sqrt{2}-\dfrac{124009}{126}\\
\end{eqnarray*}