L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{8}\) et \( Y=\left(\left(-\dfrac{17}{8}\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{8}+\left(0\right)\sqrt{18}\right)\) . 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(-\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{8}\right)+\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{8}+\left(0\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{87}{4}\right)\sqrt{2}\right)+\dfrac{27}{4}-8+\left(-21\right)\sqrt{2}\right)+\left(\left(\left(-\dfrac{85}{8}\right)\sqrt{2}\right)-\left(\left(20\right)\sqrt{2}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{87}{4}\right)\sqrt{2}\right)+\dfrac{27}{4}-8+\left(-21\right)\sqrt{2}+\left(\left(-\dfrac{85}{8}\right)\sqrt{2}\right)-\left(\left(20\right)\sqrt{2}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{679}{8}\right)\sqrt{2}-\dfrac{5}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{8}+\left(0\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{87}{4}\right)\sqrt{2}\right)+\dfrac{27}{4}-8+\left(-21\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{85}{8}\right)\sqrt{2}\right)-\left(\left(20\right)\sqrt{2}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{171}{4}\right)\sqrt{2}-\dfrac{5}{4}\right)-\left(\left(-\dfrac{337}{8}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{171}{4}\right)\sqrt{2}-\dfrac{5}{4}+\left(\dfrac{337}{8}\right)\sqrt{2}\\
&=&\left(-\dfrac{5}{8}\right)\sqrt{2}-\dfrac{5}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{29}{4}\right)\sqrt{18}\right)-\left(\left(-\dfrac{27}{8}\right)\sqrt{4}\right)-\left(\left(4\right)\sqrt{4}\right)+\left(-\dfrac{21}{2}\right)\sqrt{8}\right)\times\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{50}\right)-\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{23}{4}\right)\sqrt{8}+\left(0\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{87}{4}\right)\sqrt{2}\right)+\dfrac{27}{4}-8+\left(-21\right)\sqrt{2}\right)\times\left(\left(\left(-\dfrac{85}{8}\right)\sqrt{2}\right)-\left(\left(20\right)\sqrt{2}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{171}{4}\right)\sqrt{2}-\dfrac{5}{4}\right)\left(\left(-\dfrac{337}{8}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{57627}{32}\right)\sqrt{4}+\left(\dfrac{1685}{32}\right)\sqrt{2}\\
&=&\dfrac{57627}{16}+\left(\dfrac{1685}{32}\right)\sqrt{2}\\
\end{eqnarray*}