L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(7\right)\sqrt{20}+\left(2\right)\sqrt{20}\right)+\dfrac{55}{3}-\dfrac{14}{5}\) et \( Y=\left(\left(\left(\dfrac{59}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{20}+\left(-\dfrac{47}{5}\right)\sqrt{20}+\left(-2\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{125}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(7\right)\sqrt{20}+\left(2\right)\sqrt{20}\right)+\dfrac{55}{3}-\dfrac{14}{5}\right)+\left(\left(\left(\left(\dfrac{59}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{20}+\left(-\dfrac{47}{5}\right)\sqrt{20}+\left(-2\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(14\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)+\dfrac{55}{3}-\dfrac{14}{5}\right)+\left(\left(\dfrac{295}{4}-\left(\left(\dfrac{27}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{5}+\left(-\dfrac{94}{5}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\right)\\
&=&\left(\left(14\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)+\dfrac{55}{3}-\dfrac{14}{5}+\left(\dfrac{295}{4}-\left(\left(\dfrac{27}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{5}+\left(-\dfrac{94}{5}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\\
&=&\left(-\dfrac{1201}{30}\right)\sqrt{5}+\dfrac{16651}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(7\right)\sqrt{20}+\left(2\right)\sqrt{20}\right)+\dfrac{55}{3}-\dfrac{14}{5}\right)-\left(\left(\left(\left(\dfrac{59}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{20}+\left(-\dfrac{47}{5}\right)\sqrt{20}+\left(-2\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(14\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)+\dfrac{55}{3}-\dfrac{14}{5}\right)-\left(\left(\dfrac{295}{4}-\left(\left(\dfrac{27}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{5}+\left(-\dfrac{94}{5}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\right)\\
&=&\left(\left(18\right)\sqrt{5}+\dfrac{233}{15}\right)-\left(\dfrac{2771}{36}+\left(-\dfrac{1741}{30}\right)\sqrt{5}\right)\\
&=&\left(18\right)\sqrt{5}+\dfrac{233}{15}+-\dfrac{2771}{36}+\left(\dfrac{1741}{30}\right)\sqrt{5}\\
&=&\left(\dfrac{2281}{30}\right)\sqrt{5}-\dfrac{11059}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(7\right)\sqrt{20}+\left(2\right)\sqrt{20}\right)+\dfrac{55}{3}-\dfrac{14}{5}\right)\times\left(\left(\left(\left(\dfrac{59}{4}\right)\sqrt{25}\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{45}\right)-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{20}+\left(-\dfrac{47}{5}\right)\sqrt{20}+\left(-2\right)\sqrt{125}\right)-\left(\left(2\right)\sqrt{125}\right)\right)\\
&=&\left(\left(\left(14\right)\sqrt{5}+\left(4\right)\sqrt{5}\right)+\dfrac{55}{3}-\dfrac{14}{5}\right)\times\left(\left(\dfrac{295}{4}-\left(\left(\dfrac{27}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)+\dfrac{2}{9}\right)-\left(-3+\left(0\right)\sqrt{5}+\left(-\dfrac{94}{5}\right)\sqrt{5}+\left(-10\right)\sqrt{5}\right)-\left(\left(10\right)\sqrt{5}\right)\right)\\
&=&\left(\left(18\right)\sqrt{5}+\dfrac{233}{15}\right)\left(\dfrac{2771}{36}+\left(-\dfrac{1741}{30}\right)\sqrt{5}\right)\\
&=&\left(\dfrac{108911}{225}\right)\sqrt{5}+\left(-\dfrac{5223}{5}\right)\sqrt{25}+\dfrac{645643}{540}\\
&=&\left(\dfrac{108911}{225}\right)\sqrt{5}-\dfrac{2174777}{540}\\
\end{eqnarray*}