L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-4\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{8}\right)\) et \( Y=\left(-\dfrac{23}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{50}+\left(-6\right)\sqrt{18}+\left(\dfrac{43}{6}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(-3\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(9\right)\sqrt{18}+\left(\dfrac{23}{5}\right)\sqrt{4}\) . 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(-4\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{8}\right)\right)+\left(\left(-\dfrac{23}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{50}+\left(-6\right)\sqrt{18}+\left(\dfrac{43}{6}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(-3\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(9\right)\sqrt{18}+\left(\dfrac{23}{5}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-12\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{2}\right)\right)+\left(-\dfrac{46}{3}+\left(\dfrac{195}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(\dfrac{43}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(-6\right)\sqrt{2}+12+\left(27\right)\sqrt{2}+\dfrac{46}{5}\right)\\
&=&\left(\left(-12\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{2}\right)-\dfrac{46}{3}+\left(\dfrac{195}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(\dfrac{43}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(-6\right)\sqrt{2}+12+\left(27\right)\sqrt{2}+\dfrac{46}{5}\\
&=&\left(\dfrac{3307}{30}\right)\sqrt{2}+\dfrac{88}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-4\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{50}+\left(-6\right)\sqrt{18}+\left(\dfrac{43}{6}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(-3\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(9\right)\sqrt{18}+\left(\dfrac{23}{5}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-12\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{2}\right)\right)-\left(-\dfrac{46}{3}+\left(\dfrac{195}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(\dfrac{43}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(-6\right)\sqrt{2}+12+\left(27\right)\sqrt{2}+\dfrac{46}{5}\right)\\
&=&\left(\left(\dfrac{187}{30}\right)\sqrt{2}\right)-\left(\dfrac{88}{15}+\left(104\right)\sqrt{2}\right)\\
&=&\left(\dfrac{187}{30}\right)\sqrt{2}+-\dfrac{88}{15}+\left(-104\right)\sqrt{2}\\
&=&\left(-\dfrac{2933}{30}\right)\sqrt{2}-\dfrac{88}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-4\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{1}{5}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{6}\right)\sqrt{8}\right)\right)-\left(\left(-\dfrac{25}{4}\right)\sqrt{8}\right)\right)\times\left(\left(-\dfrac{23}{3}\right)\sqrt{4}+\left(\dfrac{39}{2}\right)\sqrt{50}+\left(-6\right)\sqrt{18}+\left(\dfrac{43}{6}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(-3\right)\sqrt{8}+\left(6\right)\sqrt{4}+\left(9\right)\sqrt{18}+\left(\dfrac{23}{5}\right)\sqrt{4}\right)\\
&=&\left(\left(\left(-12\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{3}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{25}{2}\right)\sqrt{2}\right)\right)\times\left(-\dfrac{46}{3}+\left(\dfrac{195}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(\dfrac{43}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\left(-6\right)\sqrt{2}+12+\left(27\right)\sqrt{2}+\dfrac{46}{5}\right)\\
&=&\left(\left(\dfrac{187}{30}\right)\sqrt{2}\right)\left(\dfrac{88}{15}+\left(104\right)\sqrt{2}\right)\\
&=&\left(\dfrac{8228}{225}\right)\sqrt{2}+\left(\dfrac{9724}{15}\right)\sqrt{4}\\
&=&\left(\dfrac{8228}{225}\right)\sqrt{2}+\dfrac{19448}{15}\\
\end{eqnarray*}