L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{21}{4}\right)\sqrt{20}\) et \( Y=\dfrac{17}{3}+\left(6\right)\sqrt{20}+\left(4\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{20}+\left(\dfrac{71}{8}\right)\sqrt{125}+\left(\dfrac{51}{8}\right)\sqrt{25}+\dfrac{39}{4}+9+\left(-\dfrac{23}{2}\right)\sqrt{20}+\left(\dfrac{15}{8}\right)\sqrt{45}+\left(-\dfrac{37}{9}\right)\sqrt{45}+\left(\dfrac{39}{2}\right)\sqrt{45}+\left(\dfrac{59}{7}\right)\sqrt{25}+\left(\dfrac{54}{7}\right)\sqrt{45}+\dfrac{71}{2}-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)\) . 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{21}{4}\right)\sqrt{20}\right)+\left(\dfrac{17}{3}+\left(6\right)\sqrt{20}+\left(4\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{20}+\left(\dfrac{71}{8}\right)\sqrt{125}+\left(\dfrac{51}{8}\right)\sqrt{25}+\dfrac{39}{4}+9+\left(-\dfrac{23}{2}\right)\sqrt{20}+\left(\dfrac{15}{8}\right)\sqrt{45}+\left(-\dfrac{37}{9}\right)\sqrt{45}+\left(\dfrac{39}{2}\right)\sqrt{45}+\left(\dfrac{59}{7}\right)\sqrt{25}+\left(\dfrac{54}{7}\right)\sqrt{45}+\dfrac{71}{2}-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{21}{2}\right)\sqrt{5}\right)+\left(\dfrac{17}{3}+\left(12\right)\sqrt{5}+20+\left(-\dfrac{4}{3}\right)\sqrt{5}+\left(\dfrac{355}{8}\right)\sqrt{5}+\dfrac{255}{8}+\dfrac{39}{4}+9+\left(-23\right)\sqrt{5}+\left(\dfrac{45}{8}\right)\sqrt{5}+\left(-\dfrac{37}{3}\right)\sqrt{5}+\left(\dfrac{117}{2}\right)\sqrt{5}+\dfrac{295}{7}+\left(\dfrac{162}{7}\right)\sqrt{5}+\dfrac{71}{2}-\dfrac{85}{3}\right)\\
&=&\left(\dfrac{21}{2}\right)\sqrt{5}+\dfrac{17}{3}+\left(12\right)\sqrt{5}+20+\left(-\dfrac{4}{3}\right)\sqrt{5}+\left(\dfrac{355}{8}\right)\sqrt{5}+\dfrac{255}{8}+\dfrac{39}{4}+9+\left(-23\right)\sqrt{5}+\left(\dfrac{45}{8}\right)\sqrt{5}+\left(-\dfrac{37}{3}\right)\sqrt{5}+\left(\dfrac{117}{2}\right)\sqrt{5}+\dfrac{295}{7}+\left(\dfrac{162}{7}\right)\sqrt{5}+\dfrac{71}{2}-\dfrac{85}{3}\\
&=&\left(\dfrac{2467}{21}\right)\sqrt{5}+\dfrac{21101}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{21}{4}\right)\sqrt{20}\right)-\left(\dfrac{17}{3}+\left(6\right)\sqrt{20}+\left(4\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{20}+\left(\dfrac{71}{8}\right)\sqrt{125}+\left(\dfrac{51}{8}\right)\sqrt{25}+\dfrac{39}{4}+9+\left(-\dfrac{23}{2}\right)\sqrt{20}+\left(\dfrac{15}{8}\right)\sqrt{45}+\left(-\dfrac{37}{9}\right)\sqrt{45}+\left(\dfrac{39}{2}\right)\sqrt{45}+\left(\dfrac{59}{7}\right)\sqrt{25}+\left(\dfrac{54}{7}\right)\sqrt{45}+\dfrac{71}{2}-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{21}{2}\right)\sqrt{5}\right)-\left(\dfrac{17}{3}+\left(12\right)\sqrt{5}+20+\left(-\dfrac{4}{3}\right)\sqrt{5}+\left(\dfrac{355}{8}\right)\sqrt{5}+\dfrac{255}{8}+\dfrac{39}{4}+9+\left(-23\right)\sqrt{5}+\left(\dfrac{45}{8}\right)\sqrt{5}+\left(-\dfrac{37}{3}\right)\sqrt{5}+\left(\dfrac{117}{2}\right)\sqrt{5}+\dfrac{295}{7}+\left(\dfrac{162}{7}\right)\sqrt{5}+\dfrac{71}{2}-\dfrac{85}{3}\right)\\
&=&\left(\left(\dfrac{21}{2}\right)\sqrt{5}\right)-\left(\dfrac{21101}{168}+\left(\dfrac{4493}{42}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{21}{2}\right)\sqrt{5}+-\dfrac{21101}{168}+\left(-\dfrac{4493}{42}\right)\sqrt{5}\\
&=&\left(-\dfrac{2026}{21}\right)\sqrt{5}-\dfrac{21101}{168}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{21}{4}\right)\sqrt{20}\right)\times\left(\dfrac{17}{3}+\left(6\right)\sqrt{20}+\left(4\right)\sqrt{25}+\left(-\dfrac{2}{3}\right)\sqrt{20}+\left(\dfrac{71}{8}\right)\sqrt{125}+\left(\dfrac{51}{8}\right)\sqrt{25}+\dfrac{39}{4}+9+\left(-\dfrac{23}{2}\right)\sqrt{20}+\left(\dfrac{15}{8}\right)\sqrt{45}+\left(-\dfrac{37}{9}\right)\sqrt{45}+\left(\dfrac{39}{2}\right)\sqrt{45}+\left(\dfrac{59}{7}\right)\sqrt{25}+\left(\dfrac{54}{7}\right)\sqrt{45}+\dfrac{71}{2}-\left(\left(\dfrac{17}{3}\right)\sqrt{25}\right)\right)\\
&=&\left(\left(\dfrac{21}{2}\right)\sqrt{5}\right)\times\left(\dfrac{17}{3}+\left(12\right)\sqrt{5}+20+\left(-\dfrac{4}{3}\right)\sqrt{5}+\left(\dfrac{355}{8}\right)\sqrt{5}+\dfrac{255}{8}+\dfrac{39}{4}+9+\left(-23\right)\sqrt{5}+\left(\dfrac{45}{8}\right)\sqrt{5}+\left(-\dfrac{37}{3}\right)\sqrt{5}+\left(\dfrac{117}{2}\right)\sqrt{5}+\dfrac{295}{7}+\left(\dfrac{162}{7}\right)\sqrt{5}+\dfrac{71}{2}-\dfrac{85}{3}\right)\\
&=&\left(\left(\dfrac{21}{2}\right)\sqrt{5}\right)\left(\dfrac{21101}{168}+\left(\dfrac{4493}{42}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{21101}{16}\right)\sqrt{5}+\left(\dfrac{4493}{4}\right)\sqrt{25}\\
&=&\left(\dfrac{21101}{16}\right)\sqrt{5}+\dfrac{22465}{4}\\
\end{eqnarray*}