L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{75}{4}\right)\sqrt{18}+\left(-\dfrac{13}{8}\right)\sqrt{50}+\left(-\dfrac{31}{6}\right)\sqrt{4}+\dfrac{68}{5}+\left(-\dfrac{66}{5}\right)\sqrt{18}+\left(\dfrac{31}{6}\right)\sqrt{4}+6+\left(-5\right)\sqrt{50}+\left(\left(-\dfrac{29}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{61}{5}\right)\sqrt{8}\right)\) et \( Y=\left(-6\right)\sqrt{4}\) . 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{75}{4}\right)\sqrt{18}+\left(-\dfrac{13}{8}\right)\sqrt{50}+\left(-\dfrac{31}{6}\right)\sqrt{4}+\dfrac{68}{5}+\left(-\dfrac{66}{5}\right)\sqrt{18}+\left(\dfrac{31}{6}\right)\sqrt{4}+6+\left(-5\right)\sqrt{50}+\left(\left(-\dfrac{29}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{61}{5}\right)\sqrt{8}\right)\right)+\left(\left(-6\right)\sqrt{4}\right)\\
&=&\left(\left(-\dfrac{225}{4}\right)\sqrt{2}+\left(-\dfrac{65}{8}\right)\sqrt{2}-\dfrac{31}{3}+\dfrac{68}{5}+\left(-\dfrac{198}{5}\right)\sqrt{2}+\dfrac{31}{3}+6+\left(-25\right)\sqrt{2}+\left(\left(-\dfrac{29}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{122}{5}\right)\sqrt{2}\right)\right)+\left(-12\right)\\
&=&\left(-\dfrac{225}{4}\right)\sqrt{2}+\left(-\dfrac{65}{8}\right)\sqrt{2}-\dfrac{31}{3}+\dfrac{68}{5}+\left(-\dfrac{198}{5}\right)\sqrt{2}+\dfrac{31}{3}+6+\left(-25\right)\sqrt{2}+\left(\left(-\dfrac{29}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{122}{5}\right)\sqrt{2}\right)-12\\
&=&\left(-\dfrac{1973}{12}\right)\sqrt{2}+\dfrac{38}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{75}{4}\right)\sqrt{18}+\left(-\dfrac{13}{8}\right)\sqrt{50}+\left(-\dfrac{31}{6}\right)\sqrt{4}+\dfrac{68}{5}+\left(-\dfrac{66}{5}\right)\sqrt{18}+\left(\dfrac{31}{6}\right)\sqrt{4}+6+\left(-5\right)\sqrt{50}+\left(\left(-\dfrac{29}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{61}{5}\right)\sqrt{8}\right)\right)-\left(\left(-6\right)\sqrt{4}\right)\\
&=&\left(\left(-\dfrac{225}{4}\right)\sqrt{2}+\left(-\dfrac{65}{8}\right)\sqrt{2}-\dfrac{31}{3}+\dfrac{68}{5}+\left(-\dfrac{198}{5}\right)\sqrt{2}+\dfrac{31}{3}+6+\left(-25\right)\sqrt{2}+\left(\left(-\dfrac{29}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{122}{5}\right)\sqrt{2}\right)\right)-\left(-12\right)\\
&=&\left(\left(-\dfrac{1973}{12}\right)\sqrt{2}+\dfrac{98}{5}\right)-\left(-12\right)\\
&=&\left(-\dfrac{1973}{12}\right)\sqrt{2}+\dfrac{98}{5}+12\\
&=&\left(-\dfrac{1973}{12}\right)\sqrt{2}+\dfrac{158}{5}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{75}{4}\right)\sqrt{18}+\left(-\dfrac{13}{8}\right)\sqrt{50}+\left(-\dfrac{31}{6}\right)\sqrt{4}+\dfrac{68}{5}+\left(-\dfrac{66}{5}\right)\sqrt{18}+\left(\dfrac{31}{6}\right)\sqrt{4}+6+\left(-5\right)\sqrt{50}+\left(\left(-\dfrac{29}{6}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{6}\right)\sqrt{8}\right)-\left(\left(-1\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{50}\right)-\left(\left(\dfrac{61}{5}\right)\sqrt{8}\right)\right)\times\left(\left(-6\right)\sqrt{4}\right)\\
&=&\left(\left(-\dfrac{225}{4}\right)\sqrt{2}+\left(-\dfrac{65}{8}\right)\sqrt{2}-\dfrac{31}{3}+\dfrac{68}{5}+\left(-\dfrac{198}{5}\right)\sqrt{2}+\dfrac{31}{3}+6+\left(-25\right)\sqrt{2}+\left(\left(-\dfrac{29}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{29}{3}\right)\sqrt{2}\right)-\left(\left(-5\right)\sqrt{2}\right)-\left(\left(-\dfrac{65}{8}\right)\sqrt{2}\right)-\left(\left(\dfrac{122}{5}\right)\sqrt{2}\right)\right)\times\left(-12\right)\\
&=&\left(\left(-\dfrac{1973}{12}\right)\sqrt{2}+\dfrac{98}{5}\right)\left(-12\right)\\
&=&\left(1973\right)\sqrt{2}-\dfrac{1176}{5}\\
&=&\left(1973\right)\sqrt{2}-\dfrac{1176}{5}\\
\end{eqnarray*}