L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{2}{3}\right)\sqrt{4}\right)\) et \( Y=-3-\left(\left(\dfrac{27}{7}\right)\sqrt{8}+\left(7\right)\sqrt{18}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\) . 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{1}{2}+\dfrac{1}{6}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{2}{3}\right)\sqrt{4}\right)\right)+\left(-3-\left(\left(\dfrac{27}{7}\right)\sqrt{8}+\left(7\right)\sqrt{18}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\right)\\
&=&\left(\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)+\dfrac{27}{2}-\dfrac{4}{3}\right)+\left(-3-\left(\left(\dfrac{54}{7}\right)\sqrt{2}+\left(21\right)\sqrt{2}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\right)\\
&=&\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)+\dfrac{27}{2}-\dfrac{4}{3}-3-\left(\left(\dfrac{54}{7}\right)\sqrt{2}+\left(21\right)\sqrt{2}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\\
&=&\dfrac{371}{30}+\left(-\dfrac{201}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{2}{3}\right)\sqrt{4}\right)\right)-\left(-3-\left(\left(\dfrac{27}{7}\right)\sqrt{8}+\left(7\right)\sqrt{18}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\right)\\
&=&\left(\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)+\dfrac{27}{2}-\dfrac{4}{3}\right)-\left(-3-\left(\left(\dfrac{54}{7}\right)\sqrt{2}+\left(21\right)\sqrt{2}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\right)\\
&=&\left(\dfrac{71}{6}\right)-\left(\dfrac{8}{15}+\left(-\dfrac{201}{7}\right)\sqrt{2}\right)\\
&=&\dfrac{71}{6}+-\dfrac{8}{15}+\left(\dfrac{201}{7}\right)\sqrt{2}\\
&=&\dfrac{113}{10}+\left(\dfrac{201}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)-\left(\left(-\dfrac{27}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{2}{3}\right)\sqrt{4}\right)\right)\times\left(-3-\left(\left(\dfrac{27}{7}\right)\sqrt{8}+\left(7\right)\sqrt{18}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\right)\\
&=&\left(\left(-\dfrac{1}{2}+\dfrac{1}{6}\right)+\dfrac{27}{2}-\dfrac{4}{3}\right)\times\left(-3-\left(\left(\dfrac{54}{7}\right)\sqrt{2}+\left(21\right)\sqrt{2}\right)-\left(\dfrac{49}{5}-\dfrac{40}{3}\right)\right)\\
&=&\left(\dfrac{71}{6}\right)\left(\dfrac{8}{15}+\left(-\dfrac{201}{7}\right)\sqrt{2}\right)\\
&=&\dfrac{284}{45}+\left(-\dfrac{4757}{14}\right)\sqrt{2}\\
&=&\dfrac{284}{45}+\left(-\dfrac{4757}{14}\right)\sqrt{2}\\
\end{eqnarray*}