L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-1\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{175}+0\right)\) et \( Y=-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(\dfrac{2}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{49}\right)+\left(-\dfrac{19}{3}\right)\sqrt{175}+\dfrac{22}{9}+\left(\dfrac{51}{8}\right)\sqrt{63}-7+\left(\dfrac{7}{2}\right)\sqrt{49}+\left(-\dfrac{66}{7}\right)\sqrt{28}+\left(\dfrac{17}{2}\right)\sqrt{63}+\left(\dfrac{16}{7}\right)\sqrt{63}+\left(-\dfrac{17}{7}\right)\sqrt{28}+\left(-\dfrac{25}{2}\right)\sqrt{49}+\left(-\dfrac{25}{2}\right)\sqrt{49}-7\) . 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(-1\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{175}+0\right)\right)+\left(-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(\dfrac{2}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{49}\right)+\left(-\dfrac{19}{3}\right)\sqrt{175}+\dfrac{22}{9}+\left(\dfrac{51}{8}\right)\sqrt{63}-7+\left(\dfrac{7}{2}\right)\sqrt{49}+\left(-\dfrac{66}{7}\right)\sqrt{28}+\left(\dfrac{17}{2}\right)\sqrt{63}+\left(\dfrac{16}{7}\right)\sqrt{63}+\left(-\dfrac{17}{7}\right)\sqrt{28}+\left(-\dfrac{25}{2}\right)\sqrt{49}+\left(-\dfrac{25}{2}\right)\sqrt{49}-7\right)\\
&=&\left(\left(\left(-3\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)-\left(-\dfrac{406}{5}+\left(-\dfrac{235}{9}\right)\sqrt{7}+0\right)\right)+\left(-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(2\right)\sqrt{7}\right)-\dfrac{49}{2}+\left(-\dfrac{95}{3}\right)\sqrt{7}+\dfrac{22}{9}+\left(\dfrac{153}{8}\right)\sqrt{7}-7+\dfrac{49}{2}+\left(-\dfrac{132}{7}\right)\sqrt{7}+\left(\dfrac{51}{2}\right)\sqrt{7}+\left(\dfrac{48}{7}\right)\sqrt{7}+\left(-\dfrac{34}{7}\right)\sqrt{7}-\dfrac{175}{2}-\dfrac{175}{2}-7\right)\\
&=&\left(\left(-3\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)-\left(-\dfrac{406}{5}+\left(-\dfrac{235}{9}\right)\sqrt{7}+0\right)-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(2\right)\sqrt{7}\right)-\dfrac{49}{2}+\left(-\dfrac{95}{3}\right)\sqrt{7}+\dfrac{22}{9}+\left(\dfrac{153}{8}\right)\sqrt{7}-7+\dfrac{49}{2}+\left(-\dfrac{132}{7}\right)\sqrt{7}+\left(\dfrac{51}{2}\right)\sqrt{7}+\left(\dfrac{48}{7}\right)\sqrt{7}+\left(-\dfrac{34}{7}\right)\sqrt{7}-\dfrac{175}{2}-\dfrac{175}{2}-7\\
&=&\left(\dfrac{3635}{504}\right)\sqrt{7}-\dfrac{1811}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-1\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{175}+0\right)\right)-\left(-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(\dfrac{2}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{49}\right)+\left(-\dfrac{19}{3}\right)\sqrt{175}+\dfrac{22}{9}+\left(\dfrac{51}{8}\right)\sqrt{63}-7+\left(\dfrac{7}{2}\right)\sqrt{49}+\left(-\dfrac{66}{7}\right)\sqrt{28}+\left(\dfrac{17}{2}\right)\sqrt{63}+\left(\dfrac{16}{7}\right)\sqrt{63}+\left(-\dfrac{17}{7}\right)\sqrt{28}+\left(-\dfrac{25}{2}\right)\sqrt{49}+\left(-\dfrac{25}{2}\right)\sqrt{49}-7\right)\\
&=&\left(\left(\left(-3\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)-\left(-\dfrac{406}{5}+\left(-\dfrac{235}{9}\right)\sqrt{7}+0\right)\right)-\left(-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(2\right)\sqrt{7}\right)-\dfrac{49}{2}+\left(-\dfrac{95}{3}\right)\sqrt{7}+\dfrac{22}{9}+\left(\dfrac{153}{8}\right)\sqrt{7}-7+\dfrac{49}{2}+\left(-\dfrac{132}{7}\right)\sqrt{7}+\left(\dfrac{51}{2}\right)\sqrt{7}+\left(\dfrac{48}{7}\right)\sqrt{7}+\left(-\dfrac{34}{7}\right)\sqrt{7}-\dfrac{175}{2}-\dfrac{175}{2}-7\right)\\
&=&\left(\left(\dfrac{118}{9}\right)\sqrt{7}+\dfrac{406}{5}\right)-\left(-\dfrac{687}{4}+\left(-\dfrac{991}{168}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{118}{9}\right)\sqrt{7}+\dfrac{406}{5}+\dfrac{687}{4}+\left(\dfrac{991}{168}\right)\sqrt{7}\\
&=&\left(\dfrac{9581}{504}\right)\sqrt{7}+\dfrac{5059}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-1\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)-\left(\left(-\dfrac{58}{5}\right)\sqrt{49}+\left(-\dfrac{47}{9}\right)\sqrt{175}+0\right)\right)\times\left(-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(\dfrac{2}{5}\right)\sqrt{175}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{49}\right)+\left(-\dfrac{19}{3}\right)\sqrt{175}+\dfrac{22}{9}+\left(\dfrac{51}{8}\right)\sqrt{63}-7+\left(\dfrac{7}{2}\right)\sqrt{49}+\left(-\dfrac{66}{7}\right)\sqrt{28}+\left(\dfrac{17}{2}\right)\sqrt{63}+\left(\dfrac{16}{7}\right)\sqrt{63}+\left(-\dfrac{17}{7}\right)\sqrt{28}+\left(-\dfrac{25}{2}\right)\sqrt{49}+\left(-\dfrac{25}{2}\right)\sqrt{49}-7\right)\\
&=&\left(\left(\left(-3\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)-\left(-\dfrac{406}{5}+\left(-\dfrac{235}{9}\right)\sqrt{7}+0\right)\right)\times\left(-\dfrac{4}{9}+\dfrac{61}{4}-\left(\left(2\right)\sqrt{7}\right)-\dfrac{49}{2}+\left(-\dfrac{95}{3}\right)\sqrt{7}+\dfrac{22}{9}+\left(\dfrac{153}{8}\right)\sqrt{7}-7+\dfrac{49}{2}+\left(-\dfrac{132}{7}\right)\sqrt{7}+\left(\dfrac{51}{2}\right)\sqrt{7}+\left(\dfrac{48}{7}\right)\sqrt{7}+\left(-\dfrac{34}{7}\right)\sqrt{7}-\dfrac{175}{2}-\dfrac{175}{2}-7\right)\\
&=&\left(\left(\dfrac{118}{9}\right)\sqrt{7}+\dfrac{406}{5}\right)\left(-\dfrac{687}{4}+\left(-\dfrac{991}{168}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{163849}{60}\right)\sqrt{7}+\left(-\dfrac{58469}{756}\right)\sqrt{49}-\dfrac{139461}{10}\\
&=&\left(-\dfrac{163849}{60}\right)\sqrt{7}-\dfrac{7823239}{540}\\
\end{eqnarray*}