L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-5\right)\sqrt{18}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{50}\right)-\left(\left(-7\right)\sqrt{18}+\left(-\dfrac{13}{5}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{18}+\left(-\dfrac{17}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}\right)\) et \( Y=\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{69}{7}\right)\sqrt{8}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{8}\right)+\dfrac{62}{9}-\left(\left(\dfrac{13}{2}\right)\sqrt{8}\right)\right)\) . 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(-5\right)\sqrt{18}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{50}\right)-\left(\left(-7\right)\sqrt{18}+\left(-\dfrac{13}{5}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{18}+\left(-\dfrac{17}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}\right)\right)+\left(\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{69}{7}\right)\sqrt{8}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{8}\right)+\dfrac{62}{9}-\left(\left(\dfrac{13}{2}\right)\sqrt{8}\right)\right)\right)\\
&=&\left(\left(\left(-15\right)\sqrt{2}\right)-\left(\left(\dfrac{245}{3}\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}+\left(-13\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(34\right)\sqrt{2}\right)-\left(\left(19\right)\sqrt{2}+\left(-\dfrac{34}{5}\right)\sqrt{2}+0\right)\right)+\left(\left(-\dfrac{15}{2}-\left(\left(-\dfrac{126}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{138}{7}\right)\sqrt{2}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{2}\right)+\dfrac{62}{9}-\left(\left(13\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-15\right)\sqrt{2}\right)-\left(\left(\dfrac{245}{3}\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}+\left(-13\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(34\right)\sqrt{2}\right)-\left(\left(19\right)\sqrt{2}+\left(-\dfrac{34}{5}\right)\sqrt{2}+0\right)+\left(-\dfrac{15}{2}-\left(\left(-\dfrac{126}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{138}{7}\right)\sqrt{2}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{2}\right)+\dfrac{62}{9}-\left(\left(13\right)\sqrt{2}\right)\right)\\
&=&\left(-\dfrac{3715}{42}\right)\sqrt{2}+\dfrac{2053}{210}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-5\right)\sqrt{18}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{50}\right)-\left(\left(-7\right)\sqrt{18}+\left(-\dfrac{13}{5}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{18}+\left(-\dfrac{17}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}\right)\right)-\left(\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{69}{7}\right)\sqrt{8}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{8}\right)+\dfrac{62}{9}-\left(\left(\dfrac{13}{2}\right)\sqrt{8}\right)\right)\right)\\
&=&\left(\left(\left(-15\right)\sqrt{2}\right)-\left(\left(\dfrac{245}{3}\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}+\left(-13\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(34\right)\sqrt{2}\right)-\left(\left(19\right)\sqrt{2}+\left(-\dfrac{34}{5}\right)\sqrt{2}+0\right)\right)-\left(\left(-\dfrac{15}{2}-\left(\left(-\dfrac{126}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{138}{7}\right)\sqrt{2}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{2}\right)+\dfrac{62}{9}-\left(\left(13\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-\dfrac{4391}{30}\right)\sqrt{2}+0\right)-\left(\dfrac{2053}{210}+\left(\dfrac{2027}{35}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{4391}{30}\right)\sqrt{2}+0+-\dfrac{2053}{210}+\left(-\dfrac{2027}{35}\right)\sqrt{2}\\
&=&\left(-\dfrac{42899}{210}\right)\sqrt{2}-\dfrac{2053}{210}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-5\right)\sqrt{18}\right)-\left(\left(\dfrac{49}{3}\right)\sqrt{50}\right)-\left(\left(-7\right)\sqrt{18}+\left(-\dfrac{13}{5}\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(\dfrac{34}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{19}{3}\right)\sqrt{18}+\left(-\dfrac{17}{5}\right)\sqrt{8}+\left(0\right)\sqrt{4}\right)\right)\times\left(\left(\left(\left(-\dfrac{15}{4}\right)\sqrt{4}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{69}{7}\right)\sqrt{8}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{8}\right)+\dfrac{62}{9}-\left(\left(\dfrac{13}{2}\right)\sqrt{8}\right)\right)\right)\\
&=&\left(\left(\left(-15\right)\sqrt{2}\right)-\left(\left(\dfrac{245}{3}\right)\sqrt{2}\right)-\left(\left(-21\right)\sqrt{2}+\left(-13\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(34\right)\sqrt{2}\right)-\left(\left(19\right)\sqrt{2}+\left(-\dfrac{34}{5}\right)\sqrt{2}+0\right)\right)\times\left(\left(-\dfrac{15}{2}-\left(\left(-\dfrac{126}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{138}{7}\right)\sqrt{2}\right)+\dfrac{2}{9}\right)-\left(-\dfrac{79}{5}-\dfrac{57}{7}-\left(\left(0\right)\sqrt{2}\right)+\dfrac{62}{9}-\left(\left(13\right)\sqrt{2}\right)\right)\right)\\
&=&\left(\left(-\dfrac{4391}{30}\right)\sqrt{2}+0\right)\left(\dfrac{2053}{210}+\left(\dfrac{2027}{35}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{9014723}{6300}\right)\sqrt{2}+\left(-\dfrac{8900557}{1050}\right)\sqrt{4}+0\\
&=&\left(-\dfrac{9014723}{6300}\right)\sqrt{2}-\dfrac{8900557}{525}\\
\end{eqnarray*}