L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(6\right)\sqrt{63}+\left(\left(\dfrac{73}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{65}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)+\left(\dfrac{25}{9}\right)\sqrt{49}\) et \( Y=\left(-\dfrac{70}{3}\right)\sqrt{49}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(6\right)\sqrt{63}+\left(\left(\dfrac{73}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{65}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)+\left(\dfrac{25}{9}\right)\sqrt{49}\right)+\left(\left(-\dfrac{70}{3}\right)\sqrt{49}\right)\\
&=&\left(\left(18\right)\sqrt{7}+\left(\left(\dfrac{365}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{325}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{123}{8}\right)\sqrt{7}\right)-\left(\left(19\right)\sqrt{7}\right)+\dfrac{175}{9}\right)+\left(-\dfrac{490}{3}\right)\\
&=&\left(18\right)\sqrt{7}+\left(\left(\dfrac{365}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{325}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{123}{8}\right)\sqrt{7}\right)-\left(\left(19\right)\sqrt{7}\right)+\dfrac{175}{9}-\dfrac{490}{3}\\
&=&\left(\dfrac{10549}{72}\right)\sqrt{7}-\dfrac{1295}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(6\right)\sqrt{63}+\left(\left(\dfrac{73}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{65}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)+\left(\dfrac{25}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{70}{3}\right)\sqrt{49}\right)\\
&=&\left(\left(18\right)\sqrt{7}+\left(\left(\dfrac{365}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{325}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{123}{8}\right)\sqrt{7}\right)-\left(\left(19\right)\sqrt{7}\right)+\dfrac{175}{9}\right)-\left(-\dfrac{490}{3}\right)\\
&=&\left(\left(\dfrac{10549}{72}\right)\sqrt{7}+\dfrac{175}{9}\right)-\left(-\dfrac{490}{3}\right)\\
&=&\left(\dfrac{10549}{72}\right)\sqrt{7}+\dfrac{175}{9}+\dfrac{490}{3}\\
&=&\left(\dfrac{10549}{72}\right)\sqrt{7}+\dfrac{1645}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(6\right)\sqrt{63}+\left(\left(\dfrac{73}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{63}\right)-\left(\left(\dfrac{65}{9}\right)\sqrt{175}\right)-\left(\left(\dfrac{41}{8}\right)\sqrt{63}\right)-\left(\left(\dfrac{19}{2}\right)\sqrt{28}\right)+\left(\dfrac{25}{9}\right)\sqrt{49}\right)\times\left(\left(-\dfrac{70}{3}\right)\sqrt{49}\right)\\
&=&\left(\left(18\right)\sqrt{7}+\left(\left(\dfrac{365}{2}\right)\sqrt{7}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{7}\right)-\left(\left(\dfrac{325}{9}\right)\sqrt{7}\right)-\left(\left(\dfrac{123}{8}\right)\sqrt{7}\right)-\left(\left(19\right)\sqrt{7}\right)+\dfrac{175}{9}\right)\times\left(-\dfrac{490}{3}\right)\\
&=&\left(\left(\dfrac{10549}{72}\right)\sqrt{7}+\dfrac{175}{9}\right)\left(-\dfrac{490}{3}\right)\\
&=&\left(-\dfrac{2584505}{108}\right)\sqrt{7}-\dfrac{85750}{27}\\
&=&\left(-\dfrac{2584505}{108}\right)\sqrt{7}-\dfrac{85750}{27}\\
\end{eqnarray*}