L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-8\right)\sqrt{175}\) et \( Y=\left(\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)+\left(-\dfrac{38}{3}\right)\sqrt{175}+\left(-\dfrac{18}{7}\right)\sqrt{63}+\left(\dfrac{31}{6}\right)\sqrt{28}+\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-5-\left(\left(-\dfrac{17}{6}\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{63}\right)+\left(\left(-\dfrac{57}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{49}\right)+\dfrac{71}{5}-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-8\right)\sqrt{175}\right)+\left(\left(\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)+\left(-\dfrac{38}{3}\right)\sqrt{175}+\left(-\dfrac{18}{7}\right)\sqrt{63}+\left(\dfrac{31}{6}\right)\sqrt{28}+\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-5-\left(\left(-\dfrac{17}{6}\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{63}\right)+\left(\left(-\dfrac{57}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{49}\right)+\dfrac{71}{5}-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(-40\right)\sqrt{7}\right)+\left(\left(\left(-14\right)\sqrt{7}\right)+\dfrac{98}{5}+\left(-\dfrac{190}{3}\right)\sqrt{7}+\left(-\dfrac{54}{7}\right)\sqrt{7}+\left(\dfrac{31}{3}\right)\sqrt{7}+\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-5+\dfrac{119}{6}-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{399}{2}+\dfrac{133}{3}+\dfrac{71}{5}+\dfrac{98}{5}\right)\\
&=&\left(-40\right)\sqrt{7}+\left(\left(-14\right)\sqrt{7}\right)+\dfrac{98}{5}+\left(-\dfrac{190}{3}\right)\sqrt{7}+\left(-\dfrac{54}{7}\right)\sqrt{7}+\left(\dfrac{31}{3}\right)\sqrt{7}+\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-5+\dfrac{119}{6}-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{399}{2}+\dfrac{133}{3}+\dfrac{71}{5}+\dfrac{98}{5}\\
&=&\left(-\dfrac{1961}{21}\right)\sqrt{7}-\dfrac{1304}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-8\right)\sqrt{175}\right)-\left(\left(\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)+\left(-\dfrac{38}{3}\right)\sqrt{175}+\left(-\dfrac{18}{7}\right)\sqrt{63}+\left(\dfrac{31}{6}\right)\sqrt{28}+\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-5-\left(\left(-\dfrac{17}{6}\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{63}\right)+\left(\left(-\dfrac{57}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{49}\right)+\dfrac{71}{5}-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(-40\right)\sqrt{7}\right)-\left(\left(\left(-14\right)\sqrt{7}\right)+\dfrac{98}{5}+\left(-\dfrac{190}{3}\right)\sqrt{7}+\left(-\dfrac{54}{7}\right)\sqrt{7}+\left(\dfrac{31}{3}\right)\sqrt{7}+\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-5+\dfrac{119}{6}-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{399}{2}+\dfrac{133}{3}+\dfrac{71}{5}+\dfrac{98}{5}\right)\\
&=&\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{1121}{21}\right)\sqrt{7}-\dfrac{1304}{15}\right)\\
&=&\left(-40\right)\sqrt{7}+\left(\dfrac{1121}{21}\right)\sqrt{7}+\dfrac{1304}{15}\\
&=&\left(\dfrac{281}{21}\right)\sqrt{7}+\dfrac{1304}{15}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-8\right)\sqrt{175}\right)\times\left(\left(\left(-7\right)\sqrt{28}\right)-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)+\left(-\dfrac{38}{3}\right)\sqrt{175}+\left(-\dfrac{18}{7}\right)\sqrt{63}+\left(\dfrac{31}{6}\right)\sqrt{28}+\left(\left(\dfrac{23}{3}\right)\sqrt{28}\right)-5-\left(\left(-\dfrac{17}{6}\right)\sqrt{49}\right)-\left(\left(-2\right)\sqrt{63}\right)+\left(\left(-\dfrac{57}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{19}{3}\right)\sqrt{49}\right)+\dfrac{71}{5}-\left(\left(-\dfrac{14}{5}\right)\sqrt{49}\right)\right)\\
&=&\left(\left(-40\right)\sqrt{7}\right)\times\left(\left(\left(-14\right)\sqrt{7}\right)+\dfrac{98}{5}+\left(-\dfrac{190}{3}\right)\sqrt{7}+\left(-\dfrac{54}{7}\right)\sqrt{7}+\left(\dfrac{31}{3}\right)\sqrt{7}+\left(\left(\dfrac{46}{3}\right)\sqrt{7}\right)-5+\dfrac{119}{6}-\left(\left(-6\right)\sqrt{7}\right)-\dfrac{399}{2}+\dfrac{133}{3}+\dfrac{71}{5}+\dfrac{98}{5}\right)\\
&=&\left(\left(-40\right)\sqrt{7}\right)\left(\left(-\dfrac{1121}{21}\right)\sqrt{7}-\dfrac{1304}{15}\right)\\
&=&\left(\dfrac{44840}{21}\right)\sqrt{49}+\left(\dfrac{10432}{3}\right)\sqrt{7}\\
&=&\dfrac{44840}{3}+\left(\dfrac{10432}{3}\right)\sqrt{7}\\
\end{eqnarray*}