L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{31}{3}\right)\sqrt{8}\) et \( Y=\left(\left(-\dfrac{26}{7}\right)\sqrt{50}+\left(-7\right)\sqrt{18}\right)-\left(\dfrac{15}{2}-\left(\left(3\right)\sqrt{50}\right)+\dfrac{13}{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)+\left(\left(\left(-\dfrac{26}{7}\right)\sqrt{50}+\left(-7\right)\sqrt{18}\right)-\left(\dfrac{15}{2}-\left(\left(3\right)\sqrt{50}\right)+\dfrac{13}{4}\right)\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)+\left(\left(\left(-\dfrac{130}{7}\right)\sqrt{2}+\left(-21\right)\sqrt{2}\right)-\left(\dfrac{15}{2}-\left(\left(15\right)\sqrt{2}\right)+\dfrac{13}{4}\right)\right)\\
&=&\left(-\dfrac{62}{3}\right)\sqrt{2}+\left(\left(-\dfrac{130}{7}\right)\sqrt{2}+\left(-21\right)\sqrt{2}\right)-\left(\dfrac{15}{2}-\left(\left(15\right)\sqrt{2}\right)+\dfrac{13}{4}\right)\\
&=&\left(-\dfrac{950}{21}\right)\sqrt{2}-\dfrac{43}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{26}{7}\right)\sqrt{50}+\left(-7\right)\sqrt{18}\right)-\left(\dfrac{15}{2}-\left(\left(3\right)\sqrt{50}\right)+\dfrac{13}{4}\right)\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{130}{7}\right)\sqrt{2}+\left(-21\right)\sqrt{2}\right)-\left(\dfrac{15}{2}-\left(\left(15\right)\sqrt{2}\right)+\dfrac{13}{4}\right)\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{172}{7}\right)\sqrt{2}-\dfrac{43}{4}\right)\\
&=&\left(-\dfrac{62}{3}\right)\sqrt{2}+\left(\dfrac{172}{7}\right)\sqrt{2}+\dfrac{43}{4}\\
&=&\left(\dfrac{82}{21}\right)\sqrt{2}+\dfrac{43}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{31}{3}\right)\sqrt{8}\right)\times\left(\left(\left(-\dfrac{26}{7}\right)\sqrt{50}+\left(-7\right)\sqrt{18}\right)-\left(\dfrac{15}{2}-\left(\left(3\right)\sqrt{50}\right)+\dfrac{13}{4}\right)\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)\times\left(\left(\left(-\dfrac{130}{7}\right)\sqrt{2}+\left(-21\right)\sqrt{2}\right)-\left(\dfrac{15}{2}-\left(\left(15\right)\sqrt{2}\right)+\dfrac{13}{4}\right)\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)\left(\left(-\dfrac{172}{7}\right)\sqrt{2}-\dfrac{43}{4}\right)\\
&=&\left(\dfrac{10664}{21}\right)\sqrt{4}+\left(\dfrac{1333}{6}\right)\sqrt{2}\\
&=&\dfrac{21328}{21}+\left(\dfrac{1333}{6}\right)\sqrt{2}\\
\end{eqnarray*}