L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{3}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{49}+\left(\dfrac{34}{3}\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\) et \( Y=\left(\dfrac{11}{2}-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-\left(\left(0\right)\sqrt{175}\right)\right)-\left(\left(0\right)\sqrt{175}\right)+\dfrac{31}{2}\) . 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{3}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{49}+\left(\dfrac{34}{3}\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\right)+\left(\left(\dfrac{11}{2}-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-\left(\left(0\right)\sqrt{175}\right)\right)-\left(\left(0\right)\sqrt{175}\right)+\dfrac{31}{2}\right)\\
&=&\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{7}\right)-\left(\dfrac{133}{5}+\left(\dfrac{170}{3}\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)\right)+\left(\left(\dfrac{11}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(0\right)\sqrt{7}\right)+\dfrac{31}{2}\right)\\
&=&\left(\left(-\dfrac{9}{2}\right)\sqrt{7}\right)-\left(\dfrac{133}{5}+\left(\dfrac{170}{3}\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)+\left(\dfrac{11}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(0\right)\sqrt{7}\right)+\dfrac{31}{2}\\
&=&\left(-\dfrac{203}{2}\right)\sqrt{7}-\dfrac{28}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{49}+\left(\dfrac{34}{3}\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\right)-\left(\left(\dfrac{11}{2}-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-\left(\left(0\right)\sqrt{175}\right)\right)-\left(\left(0\right)\sqrt{175}\right)+\dfrac{31}{2}\right)\\
&=&\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{7}\right)-\left(\dfrac{133}{5}+\left(\dfrac{170}{3}\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{11}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(0\right)\sqrt{7}\right)+\dfrac{31}{2}\right)\\
&=&\left(\left(-\dfrac{517}{6}\right)\sqrt{7}-\dfrac{133}{5}\right)-\left(21+\left(-\dfrac{46}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{517}{6}\right)\sqrt{7}-\dfrac{133}{5}+-21+\left(\dfrac{46}{3}\right)\sqrt{7}\\
&=&\left(-\dfrac{425}{6}\right)\sqrt{7}-\dfrac{238}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{5}\right)\sqrt{49}+\left(\dfrac{34}{3}\right)\sqrt{175}+\left(5\right)\sqrt{175}\right)\right)\times\left(\left(\dfrac{11}{2}-\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-\left(\left(0\right)\sqrt{175}\right)\right)-\left(\left(0\right)\sqrt{175}\right)+\dfrac{31}{2}\right)\\
&=&\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{7}\right)-\left(\dfrac{133}{5}+\left(\dfrac{170}{3}\right)\sqrt{7}+\left(25\right)\sqrt{7}\right)\right)\times\left(\left(\dfrac{11}{2}-\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-\left(\left(0\right)\sqrt{7}\right)\right)-\left(\left(0\right)\sqrt{7}\right)+\dfrac{31}{2}\right)\\
&=&\left(\left(-\dfrac{517}{6}\right)\sqrt{7}-\dfrac{133}{5}\right)\left(21+\left(-\dfrac{46}{3}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{42049}{30}\right)\sqrt{7}+\left(\dfrac{11891}{9}\right)\sqrt{49}-\dfrac{2793}{5}\\
&=&\left(-\dfrac{42049}{30}\right)\sqrt{7}+\dfrac{391048}{45}\\
\end{eqnarray*}