L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{61}{6}\right)\sqrt{50}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{18}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{40}{7}\right)\sqrt{18}\right)\right)\) et \( Y=\left(-\dfrac{49}{6}\right)\sqrt{4}\) . 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(-\dfrac{61}{6}\right)\sqrt{50}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{18}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{40}{7}\right)\sqrt{18}\right)\right)\right)+\left(\left(-\dfrac{49}{6}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{305}{6}\right)\sqrt{2}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{2}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{120}{7}\right)\sqrt{2}\right)\right)\right)+\left(-\dfrac{49}{3}\right)\\
&=&\left(\left(-\dfrac{305}{6}\right)\sqrt{2}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{2}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{120}{7}\right)\sqrt{2}\right)\right)-\dfrac{49}{3}\\
&=&\left(-\dfrac{3197}{42}\right)\sqrt{2}+\dfrac{83}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{61}{6}\right)\sqrt{50}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{18}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{40}{7}\right)\sqrt{18}\right)\right)\right)-\left(\left(-\dfrac{49}{6}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{305}{6}\right)\sqrt{2}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{2}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{120}{7}\right)\sqrt{2}\right)\right)\right)-\left(-\dfrac{49}{3}\right)\\
&=&\left(\left(-\dfrac{3197}{42}\right)\sqrt{2}+\dfrac{328}{15}\right)-\left(-\dfrac{49}{3}\right)\\
&=&\left(-\dfrac{3197}{42}\right)\sqrt{2}+\dfrac{328}{15}+\dfrac{49}{3}\\
&=&\left(-\dfrac{3197}{42}\right)\sqrt{2}+\dfrac{191}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{61}{6}\right)\sqrt{50}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{11}{7}\right)\sqrt{18}\right)-\left(\left(-\dfrac{8}{7}\right)\sqrt{18}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{40}{7}\right)\sqrt{18}\right)\right)\right)\times\left(\left(-\dfrac{49}{6}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-\dfrac{305}{6}\right)\sqrt{2}+9+\dfrac{6}{5}\right)-\left(\left(\left(\dfrac{33}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{24}{7}\right)\sqrt{2}\right)-\dfrac{35}{3}-\left(\left(-\dfrac{120}{7}\right)\sqrt{2}\right)\right)\right)\times\left(-\dfrac{49}{3}\right)\\
&=&\left(\left(-\dfrac{3197}{42}\right)\sqrt{2}+\dfrac{328}{15}\right)\left(-\dfrac{49}{3}\right)\\
&=&\left(\dfrac{22379}{18}\right)\sqrt{2}-\dfrac{16072}{45}\\
&=&\left(\dfrac{22379}{18}\right)\sqrt{2}-\dfrac{16072}{45}\\
\end{eqnarray*}