L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\left(-\dfrac{23}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{9}\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{175}\right)+\dfrac{27}{5}\) et \( Y=\left(\dfrac{71}{9}\right)\sqrt{28}+\left(-\dfrac{32}{3}\right)\sqrt{63}+\left(-5\right)\sqrt{28}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\left(-\dfrac{23}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{9}\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{175}\right)+\dfrac{27}{5}\right)+\left(\left(\dfrac{71}{9}\right)\sqrt{28}+\left(-\dfrac{32}{3}\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)\\
&=&\left(\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}-\dfrac{161}{8}-\left(\left(-\dfrac{22}{3}\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-48\right)\sqrt{7}\right)+\dfrac{27}{5}\right)+\left(\left(\dfrac{142}{9}\right)\sqrt{7}+\left(-32\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)\\
&=&\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}-\dfrac{161}{8}-\left(\left(-\dfrac{22}{3}\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-48\right)\sqrt{7}\right)+\dfrac{27}{5}+\left(\dfrac{142}{9}\right)\sqrt{7}+\left(-32\right)\sqrt{7}+\left(-10\right)\sqrt{7}\\
&=&-\dfrac{339}{40}+\left(\dfrac{803}{18}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\left(-\dfrac{23}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{9}\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{175}\right)+\dfrac{27}{5}\right)-\left(\left(\dfrac{71}{9}\right)\sqrt{28}+\left(-\dfrac{32}{3}\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)\\
&=&\left(\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}-\dfrac{161}{8}-\left(\left(-\dfrac{22}{3}\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-48\right)\sqrt{7}\right)+\dfrac{27}{5}\right)-\left(\left(\dfrac{142}{9}\right)\sqrt{7}+\left(-32\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{339}{40}+\left(\dfrac{425}{6}\right)\sqrt{7}\right)-\left(\left(-\dfrac{236}{9}\right)\sqrt{7}\right)\\
&=&-\dfrac{339}{40}+\left(\dfrac{425}{6}\right)\sqrt{7}+\left(\dfrac{236}{9}\right)\sqrt{7}\\
&=&-\dfrac{339}{40}+\left(\dfrac{1747}{18}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(-8\right)\sqrt{63}+\left(\left(-\dfrac{23}{8}\right)\sqrt{49}\right)-\left(\left(-\dfrac{22}{9}\right)\sqrt{63}\right)-\left(\left(-9\right)\sqrt{63}\right)-\left(\left(-\dfrac{48}{5}\right)\sqrt{175}\right)+\dfrac{27}{5}\right)\times\left(\left(\dfrac{71}{9}\right)\sqrt{28}+\left(-\dfrac{32}{3}\right)\sqrt{63}+\left(-5\right)\sqrt{28}\right)\\
&=&\left(\dfrac{61}{8}-9+\dfrac{61}{8}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(-24\right)\sqrt{7}-\dfrac{161}{8}-\left(\left(-\dfrac{22}{3}\right)\sqrt{7}\right)-\left(\left(-27\right)\sqrt{7}\right)-\left(\left(-48\right)\sqrt{7}\right)+\dfrac{27}{5}\right)\times\left(\left(\dfrac{142}{9}\right)\sqrt{7}+\left(-32\right)\sqrt{7}+\left(-10\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{339}{40}+\left(\dfrac{425}{6}\right)\sqrt{7}\right)\left(\left(-\dfrac{236}{9}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{6667}{30}\right)\sqrt{7}+\left(-\dfrac{50150}{27}\right)\sqrt{49}\\
&=&\left(\dfrac{6667}{30}\right)\sqrt{7}-\dfrac{351050}{27}\\
\end{eqnarray*}