L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{53}{8}\right)\sqrt{50}\) et \( Y=\left(\left(\dfrac{35}{3}\right)\sqrt{4}+\dfrac{22}{9}+\left(-\dfrac{66}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{18}\right)+\dfrac{19}{7}\) . 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{53}{8}\right)\sqrt{50}\right)+\left(\left(\left(\dfrac{35}{3}\right)\sqrt{4}+\dfrac{22}{9}+\left(-\dfrac{66}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{18}\right)+\dfrac{19}{7}\right)\\
&=&\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)+\left(\left(\dfrac{70}{3}+\dfrac{22}{9}+\left(-\dfrac{330}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{183}{2}\right)\sqrt{2}\right)+\dfrac{19}{7}\right)\\
&=&\left(\dfrac{265}{8}\right)\sqrt{2}+\left(\dfrac{70}{3}+\dfrac{22}{9}+\left(-\dfrac{330}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{183}{2}\right)\sqrt{2}\right)+\dfrac{19}{7}\\
&=&\left(-\dfrac{8767}{168}\right)\sqrt{2}+\dfrac{1795}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{53}{8}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{35}{3}\right)\sqrt{4}+\dfrac{22}{9}+\left(-\dfrac{66}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{18}\right)+\dfrac{19}{7}\right)\\
&=&\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{70}{3}+\dfrac{22}{9}+\left(-\dfrac{330}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{183}{2}\right)\sqrt{2}\right)+\dfrac{19}{7}\right)\\
&=&\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)-\left(\dfrac{1795}{63}+\left(-\dfrac{3583}{42}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{265}{8}\right)\sqrt{2}+-\dfrac{1795}{63}+\left(\dfrac{3583}{42}\right)\sqrt{2}\\
&=&\left(\dfrac{19897}{168}\right)\sqrt{2}-\dfrac{1795}{63}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{53}{8}\right)\sqrt{50}\right)\times\left(\left(\left(\dfrac{35}{3}\right)\sqrt{4}+\dfrac{22}{9}+\left(-\dfrac{66}{7}\right)\sqrt{50}\right)-\left(\left(-\dfrac{80}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{61}{2}\right)\sqrt{18}\right)+\dfrac{19}{7}\right)\\
&=&\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)\times\left(\left(\dfrac{70}{3}+\dfrac{22}{9}+\left(-\dfrac{330}{7}\right)\sqrt{2}\right)-\left(\left(-\dfrac{160}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{183}{2}\right)\sqrt{2}\right)+\dfrac{19}{7}\right)\\
&=&\left(\left(\dfrac{265}{8}\right)\sqrt{2}\right)\left(\dfrac{1795}{63}+\left(-\dfrac{3583}{42}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{475675}{504}\right)\sqrt{2}+\left(-\dfrac{949495}{336}\right)\sqrt{4}\\
&=&\left(\dfrac{475675}{504}\right)\sqrt{2}-\dfrac{949495}{168}\\
\end{eqnarray*}