L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{7}{4}\right)\sqrt{28}+\left(-3\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{49}\right)-\left(\left(-7\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{175}\right)\) et \( Y=-\dfrac{19}{9}\) . 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{7}{4}\right)\sqrt{28}+\left(-3\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{49}\right)-\left(\left(-7\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{175}\right)\right)+\left(-\dfrac{19}{9}\right)\\
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{41}{3}\right)\sqrt{7}\right)+\dfrac{315}{2}-\left(\left(-14\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{175}{6}\right)\sqrt{7}\right)\right)+\left(-\dfrac{19}{9}\right)\\
&=&\left(\left(-\dfrac{7}{2}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{41}{3}\right)\sqrt{7}\right)+\dfrac{315}{2}-\left(\left(-14\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{175}{6}\right)\sqrt{7}\right)-\dfrac{19}{9}\\
&=&\left(-8\right)\sqrt{7}-\dfrac{2873}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{7}{4}\right)\sqrt{28}+\left(-3\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{49}\right)-\left(\left(-7\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{175}\right)\right)-\left(-\dfrac{19}{9}\right)\\
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{41}{3}\right)\sqrt{7}\right)+\dfrac{315}{2}-\left(\left(-14\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{175}{6}\right)\sqrt{7}\right)\right)-\left(-\dfrac{19}{9}\right)\\
&=&\left(\left(-8\right)\sqrt{7}-\dfrac{315}{2}\right)-\left(-\dfrac{19}{9}\right)\\
&=&\left(-8\right)\sqrt{7}-\dfrac{315}{2}+\dfrac{19}{9}\\
&=&\left(-8\right)\sqrt{7}-\dfrac{2797}{18}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{7}{4}\right)\sqrt{28}+\left(-3\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{41}{6}\right)\sqrt{28}\right)-\left(\left(-\dfrac{45}{2}\right)\sqrt{49}\right)-\left(\left(-7\right)\sqrt{28}\right)\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{175}\right)\right)\times\left(-\dfrac{19}{9}\right)\\
&=&\left(\left(\left(-\dfrac{7}{2}\right)\sqrt{7}+\left(-6\right)\sqrt{7}\right)-\left(\left(\left(\dfrac{41}{3}\right)\sqrt{7}\right)+\dfrac{315}{2}-\left(\left(-14\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{175}{6}\right)\sqrt{7}\right)\right)\times\left(-\dfrac{19}{9}\right)\\
&=&\left(\left(-8\right)\sqrt{7}-\dfrac{315}{2}\right)\left(-\dfrac{19}{9}\right)\\
&=&\left(\dfrac{152}{9}\right)\sqrt{7}+\dfrac{665}{2}\\
&=&\left(\dfrac{152}{9}\right)\sqrt{7}+\dfrac{665}{2}\\
\end{eqnarray*}