L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{35}{2}\right)\sqrt{18}\right)-\left(-3-4-\left(\left(\dfrac{32}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{18}\right)+\dfrac{11}{3}\right)\) et \( Y=\left(\left(-\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{50}\right)+2\right)-\left(\left(\left(\dfrac{78}{7}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-2\right)\sqrt{18}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{4}\right)-\left(\left(-8\right)\sqrt{50}\right)\right)\) . 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{35}{2}\right)\sqrt{18}\right)-\left(-3-4-\left(\left(\dfrac{32}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{18}\right)+\dfrac{11}{3}\right)\right)+\left(\left(\left(-\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{50}\right)+2\right)-\left(\left(\left(\dfrac{78}{7}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-2\right)\sqrt{18}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{4}\right)-\left(\left(-8\right)\sqrt{50}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{105}{2}\right)\sqrt{2}\right)-\left(-3-4-\dfrac{64}{7}-\left(\left(\dfrac{45}{2}\right)\sqrt{2}\right)+\dfrac{11}{3}\right)\right)+\left(-\dfrac{7}{2}-\left(-\dfrac{71}{4}+\dfrac{160}{9}-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{6}\right)\sqrt{2}\right)+2\right)-\left(\left(\left(\dfrac{156}{7}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)+\dfrac{84}{5}-\left(\left(-40\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-\dfrac{105}{2}\right)\sqrt{2}\right)-\left(-3-4-\dfrac{64}{7}-\left(\left(\dfrac{45}{2}\right)\sqrt{2}\right)+\dfrac{11}{3}\right)-\dfrac{7}{2}-\left(-\dfrac{71}{4}+\dfrac{160}{9}-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{6}\right)\sqrt{2}\right)+2\right)-\left(\left(\left(\dfrac{156}{7}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)+\dfrac{84}{5}-\left(\left(-40\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{3191}{21}\right)\sqrt{2}-\dfrac{12413}{1260}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{35}{2}\right)\sqrt{18}\right)-\left(-3-4-\left(\left(\dfrac{32}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{18}\right)+\dfrac{11}{3}\right)\right)-\left(\left(\left(-\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{50}\right)+2\right)-\left(\left(\left(\dfrac{78}{7}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-2\right)\sqrt{18}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{4}\right)-\left(\left(-8\right)\sqrt{50}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{105}{2}\right)\sqrt{2}\right)-\left(-3-4-\dfrac{64}{7}-\left(\left(\dfrac{45}{2}\right)\sqrt{2}\right)+\dfrac{11}{3}\right)\right)-\left(-\dfrac{7}{2}-\left(-\dfrac{71}{4}+\dfrac{160}{9}-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{6}\right)\sqrt{2}\right)+2\right)-\left(\left(\left(\dfrac{156}{7}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)+\dfrac{84}{5}-\left(\left(-40\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-30\right)\sqrt{2}+\dfrac{262}{21}\right)-\left(-\dfrac{4019}{180}+\left(-\dfrac{2561}{21}\right)\sqrt{2}\right)\\
&=&\left(-30\right)\sqrt{2}+\dfrac{262}{21}+\dfrac{4019}{180}+\left(\dfrac{2561}{21}\right)\sqrt{2}\\
&=&\left(\dfrac{1931}{21}\right)\sqrt{2}+\dfrac{43853}{1260}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{35}{2}\right)\sqrt{18}\right)-\left(-3-4-\left(\left(\dfrac{32}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{18}\right)+\dfrac{11}{3}\right)\right)\times\left(\left(\left(-\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{71}{8}\right)\sqrt{4}\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{50}\right)+2\right)-\left(\left(\left(\dfrac{78}{7}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-2\right)\sqrt{18}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{4}\right)-\left(\left(-8\right)\sqrt{50}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{105}{2}\right)\sqrt{2}\right)-\left(-3-4-\dfrac{64}{7}-\left(\left(\dfrac{45}{2}\right)\sqrt{2}\right)+\dfrac{11}{3}\right)\right)\times\left(-\dfrac{7}{2}-\left(-\dfrac{71}{4}+\dfrac{160}{9}-\left(\left(-\dfrac{9}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{265}{6}\right)\sqrt{2}\right)+2\right)-\left(\left(\left(\dfrac{156}{7}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-6\right)\sqrt{2}\right)+\dfrac{84}{5}-\left(\left(-40\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-30\right)\sqrt{2}+\dfrac{262}{21}\right)\left(-\dfrac{4019}{180}+\left(-\dfrac{2561}{21}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{751171}{882}\right)\sqrt{2}+\left(\dfrac{25610}{7}\right)\sqrt{4}-\dfrac{526489}{1890}\\
&=&\left(-\dfrac{751171}{882}\right)\sqrt{2}+\dfrac{13302911}{1890}\\
\end{eqnarray*}