L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{31}{5}\right)\sqrt{18}+\left(\left(\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)+\left(\dfrac{57}{2}\right)\sqrt{4}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{50}\right)\) et \( Y=7-4+\dfrac{81}{2}+\left(-\dfrac{29}{2}\right)\sqrt{18}\) . 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{31}{5}\right)\sqrt{18}+\left(\left(\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)+\left(\dfrac{57}{2}\right)\sqrt{4}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{50}\right)\right)+\left(7-4+\dfrac{81}{2}+\left(-\dfrac{29}{2}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{93}{5}\right)\sqrt{2}+\dfrac{94}{7}-\left(\left(11\right)\sqrt{2}\right)+12-\left(\left(16\right)\sqrt{2}\right)+57+\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\right)+\left(7-4+\dfrac{81}{2}+\left(-\dfrac{87}{2}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{93}{5}\right)\sqrt{2}+\dfrac{94}{7}-\left(\left(11\right)\sqrt{2}\right)+12-\left(\left(16\right)\sqrt{2}\right)+57+\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)+7-4+\dfrac{81}{2}+\left(-\dfrac{87}{2}\right)\sqrt{2}\\
&=&\left(-\dfrac{1807}{30}\right)\sqrt{2}+\dfrac{1763}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{31}{5}\right)\sqrt{18}+\left(\left(\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)+\left(\dfrac{57}{2}\right)\sqrt{4}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{50}\right)\right)-\left(7-4+\dfrac{81}{2}+\left(-\dfrac{29}{2}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{93}{5}\right)\sqrt{2}+\dfrac{94}{7}-\left(\left(11\right)\sqrt{2}\right)+12-\left(\left(16\right)\sqrt{2}\right)+57+\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\right)-\left(7-4+\dfrac{81}{2}+\left(-\dfrac{87}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{251}{15}\right)\sqrt{2}+\dfrac{577}{7}\right)-\left(\dfrac{87}{2}+\left(-\dfrac{87}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{251}{15}\right)\sqrt{2}+\dfrac{577}{7}+-\dfrac{87}{2}+\left(\dfrac{87}{2}\right)\sqrt{2}\\
&=&\left(\dfrac{803}{30}\right)\sqrt{2}+\dfrac{545}{14}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{31}{5}\right)\sqrt{18}+\left(\left(\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(8\right)\sqrt{8}\right)+\left(\dfrac{57}{2}\right)\sqrt{4}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{5}{3}\right)\sqrt{50}\right)\right)\times\left(7-4+\dfrac{81}{2}+\left(-\dfrac{29}{2}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{93}{5}\right)\sqrt{2}+\dfrac{94}{7}-\left(\left(11\right)\sqrt{2}\right)+12-\left(\left(16\right)\sqrt{2}\right)+57+\left(\left(0\right)\sqrt{2}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{2}\right)\right)\times\left(7-4+\dfrac{81}{2}+\left(-\dfrac{87}{2}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{251}{15}\right)\sqrt{2}+\dfrac{577}{7}\right)\left(\dfrac{87}{2}+\left(-\dfrac{87}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{150974}{35}\right)\sqrt{2}+\left(\dfrac{7279}{10}\right)\sqrt{4}+\dfrac{50199}{14}\\
&=&\left(-\dfrac{150974}{35}\right)\sqrt{2}+\dfrac{352901}{70}\\
\end{eqnarray*}