L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{4}{3}\right)\sqrt{28}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{65}{4}\right)\sqrt{175}\right)+6-\left(\left(\dfrac{79}{7}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{71}{6}\right)\sqrt{175}+\left(-\dfrac{74}{5}\right)\sqrt{63}+\left(-\dfrac{77}{2}\right)\sqrt{49}\right)\) et \( Y=\left(\dfrac{56}{3}\right)\sqrt{63}\) . 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{4}{3}\right)\sqrt{28}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{65}{4}\right)\sqrt{175}\right)+6-\left(\left(\dfrac{79}{7}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{71}{6}\right)\sqrt{175}+\left(-\dfrac{74}{5}\right)\sqrt{63}+\left(-\dfrac{77}{2}\right)\sqrt{49}\right)\right)+\left(\left(\dfrac{56}{3}\right)\sqrt{63}\right)\\
&=&\left(\left(\left(\dfrac{8}{3}\right)\sqrt{7}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{325}{4}\right)\sqrt{7}\right)+6-\left(\left(\dfrac{237}{7}\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{355}{6}\right)\sqrt{7}+\left(-\dfrac{222}{5}\right)\sqrt{7}-\dfrac{539}{2}\right)\right)+\left(\left(56\right)\sqrt{7}\right)\\
&=&\left(\left(\dfrac{8}{3}\right)\sqrt{7}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{325}{4}\right)\sqrt{7}\right)+6-\left(\left(\dfrac{237}{7}\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{355}{6}\right)\sqrt{7}+\left(-\dfrac{222}{5}\right)\sqrt{7}-\dfrac{539}{2}\right)+\left(56\right)\sqrt{7}\\
&=&\left(-\dfrac{489}{140}\right)\sqrt{7}+\dfrac{2687}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{4}{3}\right)\sqrt{28}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{65}{4}\right)\sqrt{175}\right)+6-\left(\left(\dfrac{79}{7}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{71}{6}\right)\sqrt{175}+\left(-\dfrac{74}{5}\right)\sqrt{63}+\left(-\dfrac{77}{2}\right)\sqrt{49}\right)\right)-\left(\left(\dfrac{56}{3}\right)\sqrt{63}\right)\\
&=&\left(\left(\left(\dfrac{8}{3}\right)\sqrt{7}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{325}{4}\right)\sqrt{7}\right)+6-\left(\left(\dfrac{237}{7}\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{355}{6}\right)\sqrt{7}+\left(-\dfrac{222}{5}\right)\sqrt{7}-\dfrac{539}{2}\right)\right)-\left(\left(56\right)\sqrt{7}\right)\\
&=&\left(\left(-\dfrac{8329}{140}\right)\sqrt{7}+\dfrac{2687}{10}\right)-\left(\left(56\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{8329}{140}\right)\sqrt{7}+\dfrac{2687}{10}+\left(-56\right)\sqrt{7}\\
&=&\left(-\dfrac{16169}{140}\right)\sqrt{7}+\dfrac{2687}{10}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{4}{3}\right)\sqrt{28}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{65}{4}\right)\sqrt{175}\right)+6-\left(\left(\dfrac{79}{7}\right)\sqrt{63}\right)\right)-\left(\left(\dfrac{71}{6}\right)\sqrt{175}+\left(-\dfrac{74}{5}\right)\sqrt{63}+\left(-\dfrac{77}{2}\right)\sqrt{49}\right)\right)\times\left(\left(\dfrac{56}{3}\right)\sqrt{63}\right)\\
&=&\left(\left(\left(\dfrac{8}{3}\right)\sqrt{7}\right)+\dfrac{26}{5}-\left(\left(\left(\dfrac{325}{4}\right)\sqrt{7}\right)+6-\left(\left(\dfrac{237}{7}\right)\sqrt{7}\right)\right)-\left(\left(\dfrac{355}{6}\right)\sqrt{7}+\left(-\dfrac{222}{5}\right)\sqrt{7}-\dfrac{539}{2}\right)\right)\times\left(\left(56\right)\sqrt{7}\right)\\
&=&\left(\left(-\dfrac{8329}{140}\right)\sqrt{7}+\dfrac{2687}{10}\right)\left(\left(56\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{16658}{5}\right)\sqrt{49}+\left(\dfrac{75236}{5}\right)\sqrt{7}\\
&=&-\dfrac{116606}{5}+\left(\dfrac{75236}{5}\right)\sqrt{7}\\
\end{eqnarray*}