L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=-\dfrac{62}{9}+\left(7\right)\sqrt{8}-\dfrac{30}{7}+\left(\dfrac{47}{2}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(\dfrac{42}{5}\right)\sqrt{4}\) et \( Y=-\dfrac{19}{2}-\left(\left(\dfrac{19}{7}\right)\sqrt{18}\right)+2+\left(-\dfrac{9}{4}\right)\sqrt{18}+\left(-\dfrac{37}{7}\right)\sqrt{4}+\dfrac{33}{2}+\left(-\dfrac{36}{5}\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(8\right)\sqrt{18}+\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(\left(9\right)\sqrt{18}\right)-\dfrac{1}{2}-\left(\left(-9\right)\sqrt{18}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(-\dfrac{62}{9}+\left(7\right)\sqrt{8}-\dfrac{30}{7}+\left(\dfrac{47}{2}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(\dfrac{42}{5}\right)\sqrt{4}\right)+\left(-\dfrac{19}{2}-\left(\left(\dfrac{19}{7}\right)\sqrt{18}\right)+2+\left(-\dfrac{9}{4}\right)\sqrt{18}+\left(-\dfrac{37}{7}\right)\sqrt{4}+\dfrac{33}{2}+\left(-\dfrac{36}{5}\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(8\right)\sqrt{18}+\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(\left(9\right)\sqrt{18}\right)-\dfrac{1}{2}-\left(\left(-9\right)\sqrt{18}\right)\right)\\
&=&\left(-\dfrac{62}{9}+\left(14\right)\sqrt{2}-\dfrac{30}{7}+\left(\dfrac{141}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\dfrac{84}{5}\right)+\left(-\dfrac{19}{2}-\left(\left(\dfrac{57}{7}\right)\sqrt{2}\right)+2+\left(-\dfrac{27}{4}\right)\sqrt{2}-\dfrac{74}{7}+\dfrac{33}{2}+\left(-36\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(24\right)\sqrt{2}+\left(-\dfrac{75}{4}\right)\sqrt{2}+\left(\left(27\right)\sqrt{2}\right)-\dfrac{1}{2}-\left(\left(-27\right)\sqrt{2}\right)\right)\\
&=&-\dfrac{62}{9}+\left(14\right)\sqrt{2}-\dfrac{30}{7}+\left(\dfrac{141}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\dfrac{84}{5}-\dfrac{19}{2}-\left(\left(\dfrac{57}{7}\right)\sqrt{2}\right)+2+\left(-\dfrac{27}{4}\right)\sqrt{2}-\dfrac{74}{7}+\dfrac{33}{2}+\left(-36\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(24\right)\sqrt{2}+\left(-\dfrac{75}{4}\right)\sqrt{2}+\left(\left(27\right)\sqrt{2}\right)-\dfrac{1}{2}-\left(\left(-27\right)\sqrt{2}\right)\\
&=&\dfrac{2239}{630}+\left(\dfrac{244}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(-\dfrac{62}{9}+\left(7\right)\sqrt{8}-\dfrac{30}{7}+\left(\dfrac{47}{2}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(\dfrac{42}{5}\right)\sqrt{4}\right)-\left(-\dfrac{19}{2}-\left(\left(\dfrac{19}{7}\right)\sqrt{18}\right)+2+\left(-\dfrac{9}{4}\right)\sqrt{18}+\left(-\dfrac{37}{7}\right)\sqrt{4}+\dfrac{33}{2}+\left(-\dfrac{36}{5}\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(8\right)\sqrt{18}+\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(\left(9\right)\sqrt{18}\right)-\dfrac{1}{2}-\left(\left(-9\right)\sqrt{18}\right)\right)\\
&=&\left(-\dfrac{62}{9}+\left(14\right)\sqrt{2}-\dfrac{30}{7}+\left(\dfrac{141}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\dfrac{84}{5}\right)-\left(-\dfrac{19}{2}-\left(\left(\dfrac{57}{7}\right)\sqrt{2}\right)+2+\left(-\dfrac{27}{4}\right)\sqrt{2}-\dfrac{74}{7}+\dfrac{33}{2}+\left(-36\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(24\right)\sqrt{2}+\left(-\dfrac{75}{4}\right)\sqrt{2}+\left(\left(27\right)\sqrt{2}\right)-\dfrac{1}{2}-\left(\left(-27\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{1772}{315}+\left(\dfrac{133}{2}\right)\sqrt{2}\right)-\left(-\dfrac{29}{14}+\left(-\dfrac{443}{14}\right)\sqrt{2}\right)\\
&=&\dfrac{1772}{315}+\left(\dfrac{133}{2}\right)\sqrt{2}+\dfrac{29}{14}+\left(\dfrac{443}{14}\right)\sqrt{2}\\
&=&\dfrac{4849}{630}+\left(\dfrac{687}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(-\dfrac{62}{9}+\left(7\right)\sqrt{8}-\dfrac{30}{7}+\left(\dfrac{47}{2}\right)\sqrt{18}+\left(-6\right)\sqrt{18}+\left(\dfrac{42}{5}\right)\sqrt{4}\right)\times\left(-\dfrac{19}{2}-\left(\left(\dfrac{19}{7}\right)\sqrt{18}\right)+2+\left(-\dfrac{9}{4}\right)\sqrt{18}+\left(-\dfrac{37}{7}\right)\sqrt{4}+\dfrac{33}{2}+\left(-\dfrac{36}{5}\right)\sqrt{50}+\left(-8\right)\sqrt{50}+\left(8\right)\sqrt{18}+\left(-\dfrac{25}{4}\right)\sqrt{18}+\left(\left(9\right)\sqrt{18}\right)-\dfrac{1}{2}-\left(\left(-9\right)\sqrt{18}\right)\right)\\
&=&\left(-\dfrac{62}{9}+\left(14\right)\sqrt{2}-\dfrac{30}{7}+\left(\dfrac{141}{2}\right)\sqrt{2}+\left(-18\right)\sqrt{2}+\dfrac{84}{5}\right)\times\left(-\dfrac{19}{2}-\left(\left(\dfrac{57}{7}\right)\sqrt{2}\right)+2+\left(-\dfrac{27}{4}\right)\sqrt{2}-\dfrac{74}{7}+\dfrac{33}{2}+\left(-36\right)\sqrt{2}+\left(-40\right)\sqrt{2}+\left(24\right)\sqrt{2}+\left(-\dfrac{75}{4}\right)\sqrt{2}+\left(\left(27\right)\sqrt{2}\right)-\dfrac{1}{2}-\left(\left(-27\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{1772}{315}+\left(\dfrac{133}{2}\right)\sqrt{2}\right)\left(-\dfrac{29}{14}+\left(-\dfrac{443}{14}\right)\sqrt{2}\right)\\
&=&-\dfrac{25694}{2205}+\left(-\dfrac{2784947}{8820}\right)\sqrt{2}+\left(-\dfrac{8417}{4}\right)\sqrt{4}\\
&=&-\dfrac{18610873}{4410}+\left(-\dfrac{2784947}{8820}\right)\sqrt{2}\\
\end{eqnarray*}