L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{77}{5}-\left(\left(-\dfrac{59}{5}\right)\sqrt{175}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)-\left(\left(\left(-8\right)\sqrt{49}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{175}\right)+\dfrac{63}{8}\right)-\left(\left(\left(\dfrac{59}{9}\right)\sqrt{49}\right)+\dfrac{34}{3}-\left(\left(9\right)\sqrt{28}\right)-\dfrac{3}{2}\right)\) et \( Y=\left(-\dfrac{17}{5}-\left(\left(\dfrac{29}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{54}{7}\right)\sqrt{175}\right)\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{28}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{77}{5}-\left(\left(-\dfrac{59}{5}\right)\sqrt{175}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)-\left(\left(\left(-8\right)\sqrt{49}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{175}\right)+\dfrac{63}{8}\right)-\left(\left(\left(\dfrac{59}{9}\right)\sqrt{49}\right)+\dfrac{34}{3}-\left(\left(9\right)\sqrt{28}\right)-\dfrac{3}{2}\right)\right)+\left(\left(-\dfrac{17}{5}-\left(\left(\dfrac{29}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{54}{7}\right)\sqrt{175}\right)\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{28}\right)\right)\\
&=&\left(\left(-\dfrac{77}{5}-\left(\left(-59\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)\right)-\left(\left(-13\right)\sqrt{7}\right)-\left(-56-\left(\left(-\dfrac{5}{3}\right)\sqrt{7}\right)+\dfrac{63}{8}\right)-\left(\dfrac{413}{9}+\dfrac{34}{3}-\left(\left(18\right)\sqrt{7}\right)-\dfrac{3}{2}\right)\right)+\left(\left(-\dfrac{17}{5}-\left(\left(\dfrac{87}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{270}{7}\right)\sqrt{7}\right)\right)+\dfrac{455}{6}-\left(\left(\dfrac{2}{9}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{77}{5}-\left(\left(-59\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)\right)-\left(\left(-13\right)\sqrt{7}\right)-\left(-56-\left(\left(-\dfrac{5}{3}\right)\sqrt{7}\right)+\dfrac{63}{8}\right)-\left(\dfrac{413}{9}+\dfrac{34}{3}-\left(\left(18\right)\sqrt{7}\right)-\dfrac{3}{2}\right)+\left(-\dfrac{17}{5}-\left(\left(\dfrac{87}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{270}{7}\right)\sqrt{7}\right)\right)+\dfrac{455}{6}-\left(\left(\dfrac{2}{9}\right)\sqrt{7}\right)\\
&=&\dfrac{17797}{360}+\left(\dfrac{40051}{252}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{77}{5}-\left(\left(-\dfrac{59}{5}\right)\sqrt{175}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)-\left(\left(\left(-8\right)\sqrt{49}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{175}\right)+\dfrac{63}{8}\right)-\left(\left(\left(\dfrac{59}{9}\right)\sqrt{49}\right)+\dfrac{34}{3}-\left(\left(9\right)\sqrt{28}\right)-\dfrac{3}{2}\right)\right)-\left(\left(-\dfrac{17}{5}-\left(\left(\dfrac{29}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{54}{7}\right)\sqrt{175}\right)\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{28}\right)\right)\\
&=&\left(\left(-\dfrac{77}{5}-\left(\left(-59\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)\right)-\left(\left(-13\right)\sqrt{7}\right)-\left(-56-\left(\left(-\dfrac{5}{3}\right)\sqrt{7}\right)+\dfrac{63}{8}\right)-\left(\dfrac{413}{9}+\dfrac{34}{3}-\left(\left(18\right)\sqrt{7}\right)-\dfrac{3}{2}\right)\right)-\left(\left(-\dfrac{17}{5}-\left(\left(\dfrac{87}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{270}{7}\right)\sqrt{7}\right)\right)+\dfrac{455}{6}-\left(\left(\dfrac{2}{9}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{8279}{360}+\left(\dfrac{427}{3}\right)\sqrt{7}\right)-\left(\dfrac{2173}{30}+\left(\dfrac{4183}{252}\right)\sqrt{7}\right)\\
&=&-\dfrac{8279}{360}+\left(\dfrac{427}{3}\right)\sqrt{7}+-\dfrac{2173}{30}+\left(-\dfrac{4183}{252}\right)\sqrt{7}\\
&=&-\dfrac{6871}{72}+\left(\dfrac{31685}{252}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{77}{5}-\left(\left(-\dfrac{59}{5}\right)\sqrt{175}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{13}{2}\right)\sqrt{28}\right)-\left(\left(\left(-8\right)\sqrt{49}\right)-\left(\left(-\dfrac{1}{3}\right)\sqrt{175}\right)+\dfrac{63}{8}\right)-\left(\left(\left(\dfrac{59}{9}\right)\sqrt{49}\right)+\dfrac{34}{3}-\left(\left(9\right)\sqrt{28}\right)-\dfrac{3}{2}\right)\right)\times\left(\left(-\dfrac{17}{5}-\left(\left(\dfrac{29}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{54}{7}\right)\sqrt{175}\right)\right)-\left(\left(-\dfrac{65}{6}\right)\sqrt{49}\right)-\left(\left(\dfrac{1}{9}\right)\sqrt{28}\right)\right)\\
&=&\left(\left(-\dfrac{77}{5}-\left(\left(-59\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)\right)-\left(\left(-13\right)\sqrt{7}\right)-\left(-56-\left(\left(-\dfrac{5}{3}\right)\sqrt{7}\right)+\dfrac{63}{8}\right)-\left(\dfrac{413}{9}+\dfrac{34}{3}-\left(\left(18\right)\sqrt{7}\right)-\dfrac{3}{2}\right)\right)\times\left(\left(-\dfrac{17}{5}-\left(\left(\dfrac{87}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{270}{7}\right)\sqrt{7}\right)\right)+\dfrac{455}{6}-\left(\left(\dfrac{2}{9}\right)\sqrt{7}\right)\right)\\
&=&\left(-\dfrac{8279}{360}+\left(\dfrac{427}{3}\right)\sqrt{7}\right)\left(\dfrac{2173}{30}+\left(\dfrac{4183}{252}\right)\sqrt{7}\right)\\
&=&-\dfrac{17990267}{10800}+\left(\dfrac{81059661990}{8164800}\right)\sqrt{7}+\left(\dfrac{255163}{108}\right)\sqrt{49}\\
&=&\dfrac{17347373964}{1166400}+\left(\dfrac{81059661990}{8164800}\right)\sqrt{7}\\
\end{eqnarray*}