L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{43}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{175}\right)-\left(\left(9\right)\sqrt{49}\right)-\dfrac{77}{2}+5+\left(-\dfrac{53}{8}\right)\sqrt{175}\) et \( Y=\dfrac{49}{9}-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{49}\right)+\dfrac{21}{5}-\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)+\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{49}\right)\) . 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{43}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{175}\right)-\left(\left(9\right)\sqrt{49}\right)-\dfrac{77}{2}+5+\left(-\dfrac{53}{8}\right)\sqrt{175}\right)+\left(\dfrac{49}{9}-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{49}\right)+\dfrac{21}{5}-\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)+\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-\dfrac{215}{3}\right)\sqrt{7}\right)+\dfrac{21}{2}-\left(\left(-\dfrac{150}{7}\right)\sqrt{7}\right)-63-\dfrac{77}{2}+5+\left(-\dfrac{265}{8}\right)\sqrt{7}\right)+\left(\dfrac{49}{9}+49-\dfrac{21}{4}+\dfrac{21}{5}-\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)+\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{125}{8}\right)\sqrt{7}\right)+39\right)\\
&=&\left(\left(-\dfrac{215}{3}\right)\sqrt{7}\right)+\dfrac{21}{2}-\left(\left(-\dfrac{150}{7}\right)\sqrt{7}\right)-63-\dfrac{77}{2}+5+\left(-\dfrac{265}{8}\right)\sqrt{7}+\dfrac{49}{9}+49-\dfrac{21}{4}+\dfrac{21}{5}-\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)+\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{125}{8}\right)\sqrt{7}\right)+39\\
&=&\left(-\dfrac{8315}{84}\right)\sqrt{7}+\dfrac{1151}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{43}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{175}\right)-\left(\left(9\right)\sqrt{49}\right)-\dfrac{77}{2}+5+\left(-\dfrac{53}{8}\right)\sqrt{175}\right)-\left(\dfrac{49}{9}-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{49}\right)+\dfrac{21}{5}-\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)+\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-\dfrac{215}{3}\right)\sqrt{7}\right)+\dfrac{21}{2}-\left(\left(-\dfrac{150}{7}\right)\sqrt{7}\right)-63-\dfrac{77}{2}+5+\left(-\dfrac{265}{8}\right)\sqrt{7}\right)-\left(\dfrac{49}{9}+49-\dfrac{21}{4}+\dfrac{21}{5}-\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)+\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{125}{8}\right)\sqrt{7}\right)+39\right)\\
&=&\left(\left(-\dfrac{14005}{168}\right)\sqrt{7}-86\right)-\left(\dfrac{16631}{180}+\left(-\dfrac{125}{8}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{14005}{168}\right)\sqrt{7}-86+-\dfrac{16631}{180}+\left(\dfrac{125}{8}\right)\sqrt{7}\\
&=&\left(-\dfrac{2845}{42}\right)\sqrt{7}-\dfrac{32111}{180}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{43}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{30}{7}\right)\sqrt{175}\right)-\left(\left(9\right)\sqrt{49}\right)-\dfrac{77}{2}+5+\left(-\dfrac{53}{8}\right)\sqrt{175}\right)\times\left(\dfrac{49}{9}-\left(\left(-7\right)\sqrt{49}\right)-\left(\left(\dfrac{3}{4}\right)\sqrt{49}\right)+\dfrac{21}{5}-\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)+\left(\left(-\dfrac{37}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{25}{8}\right)\sqrt{175}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(\left(-\dfrac{215}{3}\right)\sqrt{7}\right)+\dfrac{21}{2}-\left(\left(-\dfrac{150}{7}\right)\sqrt{7}\right)-63-\dfrac{77}{2}+5+\left(-\dfrac{265}{8}\right)\sqrt{7}\right)\times\left(\dfrac{49}{9}+49-\dfrac{21}{4}+\dfrac{21}{5}-\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)+\left(\left(-\dfrac{111}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{125}{8}\right)\sqrt{7}\right)+39\right)\\
&=&\left(\left(-\dfrac{14005}{168}\right)\sqrt{7}-86\right)\left(\dfrac{16631}{180}+\left(-\dfrac{125}{8}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{38456431}{6048}\right)\sqrt{7}+\left(\dfrac{1750625}{1344}\right)\sqrt{49}-\dfrac{715133}{90}\\
&=&\left(-\dfrac{38456431}{6048}\right)\sqrt{7}+\dfrac{3375119}{2880}\\
\end{eqnarray*}