L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{3}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{8}\right)+\left(-\dfrac{74}{3}\right)\sqrt{18}+\left(\dfrac{66}{5}\right)\sqrt{50}-\dfrac{71}{6}\) et \( Y=\left(-\dfrac{27}{8}\right)\sqrt{50}+\left(\left(-8\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{50}\right)+9\) . 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{3}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{8}\right)+\left(-\dfrac{74}{3}\right)\sqrt{18}+\left(\dfrac{66}{5}\right)\sqrt{50}-\dfrac{71}{6}\right)+\left(\left(-\dfrac{27}{8}\right)\sqrt{50}+\left(\left(-8\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{50}\right)+9\right)\\
&=&\left(-3-\left(\left(66\right)\sqrt{2}\right)-\left(\left(-17\right)\sqrt{2}\right)+\left(-74\right)\sqrt{2}+\left(66\right)\sqrt{2}-\dfrac{71}{6}\right)+\left(\left(-\dfrac{135}{8}\right)\sqrt{2}+\left(\left(-24\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)+9\right)\\
&=&-3-\left(\left(66\right)\sqrt{2}\right)-\left(\left(-17\right)\sqrt{2}\right)+\left(-74\right)\sqrt{2}+\left(66\right)\sqrt{2}-\dfrac{71}{6}+\left(-\dfrac{135}{8}\right)\sqrt{2}+\left(\left(-24\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)+9\\
&=&-\dfrac{35}{6}+\left(-\dfrac{567}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{8}\right)+\left(-\dfrac{74}{3}\right)\sqrt{18}+\left(\dfrac{66}{5}\right)\sqrt{50}-\dfrac{71}{6}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{50}+\left(\left(-8\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{50}\right)+9\right)\\
&=&\left(-3-\left(\left(66\right)\sqrt{2}\right)-\left(\left(-17\right)\sqrt{2}\right)+\left(-74\right)\sqrt{2}+\left(66\right)\sqrt{2}-\dfrac{71}{6}\right)-\left(\left(-\dfrac{135}{8}\right)\sqrt{2}+\left(\left(-24\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)+9\right)\\
&=&\left(-\dfrac{89}{6}+\left(-57\right)\sqrt{2}\right)-\left(\left(-\dfrac{111}{8}\right)\sqrt{2}+9\right)\\
&=&-\dfrac{89}{6}+\left(-57\right)\sqrt{2}+\left(\dfrac{111}{8}\right)\sqrt{2}-9\\
&=&-\dfrac{143}{6}+\left(-\dfrac{345}{8}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{66}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{17}{2}\right)\sqrt{8}\right)+\left(-\dfrac{74}{3}\right)\sqrt{18}+\left(\dfrac{66}{5}\right)\sqrt{50}-\dfrac{71}{6}\right)\times\left(\left(-\dfrac{27}{8}\right)\sqrt{50}+\left(\left(-8\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{50}\right)+9\right)\\
&=&\left(-3-\left(\left(66\right)\sqrt{2}\right)-\left(\left(-17\right)\sqrt{2}\right)+\left(-74\right)\sqrt{2}+\left(66\right)\sqrt{2}-\dfrac{71}{6}\right)\times\left(\left(-\dfrac{135}{8}\right)\sqrt{2}+\left(\left(-24\right)\sqrt{2}\right)-\left(\left(-27\right)\sqrt{2}\right)+9\right)\\
&=&\left(-\dfrac{89}{6}+\left(-57\right)\sqrt{2}\right)\left(\left(-\dfrac{111}{8}\right)\sqrt{2}+9\right)\\
&=&\left(-\dfrac{4915}{16}\right)\sqrt{2}-\dfrac{267}{2}+\left(\dfrac{6327}{8}\right)\sqrt{4}\\
&=&\left(-\dfrac{4915}{16}\right)\sqrt{2}+\dfrac{5793}{4}\\
\end{eqnarray*}