L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(1\right)\sqrt{8}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{79}{3}\right)\sqrt{50}\right)-\dfrac{33}{4}-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{18}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)\) et \( Y=\left(\left(-9\right)\sqrt{50}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{4}\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{18}\right)\) . 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(\left(-8\right)\sqrt{8}\right)-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(1\right)\sqrt{8}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{79}{3}\right)\sqrt{50}\right)-\dfrac{33}{4}-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{18}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)\right)+\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{4}\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{395}{3}\right)\sqrt{2}\right)-\dfrac{33}{4}-\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-19\right)\sqrt{2}\right)\right)+\left(\left(\left(-45\right)\sqrt{2}\right)-\left(-\dfrac{1}{2}-15\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-76\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{395}{3}\right)\sqrt{2}\right)-\dfrac{33}{4}-\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-19\right)\sqrt{2}\right)+\left(\left(-45\right)\sqrt{2}\right)-\left(-\dfrac{1}{2}-15\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-76\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{308}{3}\right)\sqrt{2}+\dfrac{35}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(1\right)\sqrt{8}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{79}{3}\right)\sqrt{50}\right)-\dfrac{33}{4}-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{18}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)\right)-\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{4}\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{395}{3}\right)\sqrt{2}\right)-\dfrac{33}{4}-\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-19\right)\sqrt{2}\right)\right)-\left(\left(\left(-45\right)\sqrt{2}\right)-\left(-\dfrac{1}{2}-15\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-76\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{401}{3}\right)\sqrt{2}+\dfrac{27}{2}\right)-\left(\left(31\right)\sqrt{2}-\dfrac{19}{4}\right)\\
&=&\left(-\dfrac{401}{3}\right)\sqrt{2}+\dfrac{27}{2}+\left(-31\right)\sqrt{2}+\dfrac{19}{4}\\
&=&\left(-\dfrac{494}{3}\right)\sqrt{2}+\dfrac{73}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-8\right)\sqrt{8}\right)-\left(\left(-7\right)\sqrt{18}\right)\right)-\left(\left(1\right)\sqrt{8}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{79}{3}\right)\sqrt{50}\right)-\dfrac{33}{4}-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{18}\right)\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{8}\right)\right)\times\left(\left(\left(-9\right)\sqrt{50}\right)-\left(\left(-\dfrac{1}{4}\right)\sqrt{4}+\left(-\dfrac{15}{2}\right)\sqrt{4}\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-\dfrac{76}{3}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(\left(-16\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}\right)\right)-\left(\left(2\right)\sqrt{2}-\dfrac{21}{4}\right)-\left(\left(\left(\dfrac{395}{3}\right)\sqrt{2}\right)-\dfrac{33}{4}-\left(\left(-19\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-19\right)\sqrt{2}\right)\right)\times\left(\left(\left(-45\right)\sqrt{2}\right)-\left(-\dfrac{1}{2}-15\right)-\left(\dfrac{61}{4}+5\right)-\left(\left(-76\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{401}{3}\right)\sqrt{2}+\dfrac{27}{2}\right)\left(\left(31\right)\sqrt{2}-\dfrac{19}{4}\right)\\
&=&\left(-\dfrac{12431}{3}\right)\sqrt{4}+\left(\dfrac{12641}{12}\right)\sqrt{2}-\dfrac{513}{8}\\
&=&-\dfrac{200435}{24}+\left(\dfrac{12641}{12}\right)\sqrt{2}\\
\end{eqnarray*}