L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=6+\left(\left(6\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)+\left(\left(-\dfrac{79}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{71}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{18}\right)-\dfrac{23}{3}\) et \( Y=\left(2\right)\sqrt{18}-1+\left(-\dfrac{26}{3}\right)\sqrt{4}+\left(\dfrac{35}{3}\right)\sqrt{8}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{29}{8}\right)\sqrt{18}+\left(-2\right)\sqrt{8}+\left(\dfrac{30}{7}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(6+\left(\left(6\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)+\left(\left(-\dfrac{79}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{71}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{18}\right)-\dfrac{23}{3}\right)+\left(\left(2\right)\sqrt{18}-1+\left(-\dfrac{26}{3}\right)\sqrt{4}+\left(\dfrac{35}{3}\right)\sqrt{8}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{29}{8}\right)\sqrt{18}+\left(-2\right)\sqrt{8}+\left(\dfrac{30}{7}\right)\sqrt{50}\right)\\
&=&\left(6+\left(\left(18\right)\sqrt{2}\right)+12-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\left(\left(-79\right)\sqrt{2}\right)-\left(\left(-\dfrac{355}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{2}\right)-\dfrac{23}{3}\right)+\left(\left(6\right)\sqrt{2}-1-\dfrac{52}{3}+\left(\dfrac{70}{3}\right)\sqrt{2}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{87}{8}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{150}{7}\right)\sqrt{2}\right)\\
&=&6+\left(\left(18\right)\sqrt{2}\right)+12-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\left(\left(-79\right)\sqrt{2}\right)-\left(\left(-\dfrac{355}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{2}\right)-\dfrac{23}{3}+\left(6\right)\sqrt{2}-1-\dfrac{52}{3}+\left(\dfrac{70}{3}\right)\sqrt{2}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{87}{8}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{150}{7}\right)\sqrt{2}\\
&=&-\dfrac{301}{12}+\left(-\dfrac{3364}{105}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(6+\left(\left(6\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)+\left(\left(-\dfrac{79}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{71}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{18}\right)-\dfrac{23}{3}\right)-\left(\left(2\right)\sqrt{18}-1+\left(-\dfrac{26}{3}\right)\sqrt{4}+\left(\dfrac{35}{3}\right)\sqrt{8}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{29}{8}\right)\sqrt{18}+\left(-2\right)\sqrt{8}+\left(\dfrac{30}{7}\right)\sqrt{50}\right)\\
&=&\left(6+\left(\left(18\right)\sqrt{2}\right)+12-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\left(\left(-79\right)\sqrt{2}\right)-\left(\left(-\dfrac{355}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{2}\right)-\dfrac{23}{3}\right)-\left(\left(6\right)\sqrt{2}-1-\dfrac{52}{3}+\left(\dfrac{70}{3}\right)\sqrt{2}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{87}{8}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{150}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{31}{3}+\left(-\dfrac{2717}{40}\right)\sqrt{2}\right)-\left(\left(\dfrac{6029}{168}\right)\sqrt{2}-\dfrac{425}{12}\right)\\
&=&\dfrac{31}{3}+\left(-\dfrac{2717}{40}\right)\sqrt{2}+\left(-\dfrac{6029}{168}\right)\sqrt{2}+\dfrac{425}{12}\\
&=&\dfrac{183}{4}+\left(-\dfrac{43601}{420}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(6+\left(\left(6\right)\sqrt{18}\right)-\left(\left(-6\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)+\left(\left(-\dfrac{79}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{71}{8}\right)\sqrt{50}\right)-\left(\left(-\dfrac{27}{5}\right)\sqrt{18}\right)-\dfrac{23}{3}\right)\times\left(\left(2\right)\sqrt{18}-1+\left(-\dfrac{26}{3}\right)\sqrt{4}+\left(\dfrac{35}{3}\right)\sqrt{8}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{29}{8}\right)\sqrt{18}+\left(-2\right)\sqrt{8}+\left(\dfrac{30}{7}\right)\sqrt{50}\right)\\
&=&\left(6+\left(\left(18\right)\sqrt{2}\right)+12-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\left(\left(-79\right)\sqrt{2}\right)-\left(\left(-\dfrac{355}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{81}{5}\right)\sqrt{2}\right)-\dfrac{23}{3}\right)\times\left(\left(6\right)\sqrt{2}-1-\dfrac{52}{3}+\left(\dfrac{70}{3}\right)\sqrt{2}-\dfrac{75}{4}+\dfrac{5}{3}+\left(-\dfrac{87}{8}\right)\sqrt{2}+\left(-4\right)\sqrt{2}+\left(\dfrac{150}{7}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{31}{3}+\left(-\dfrac{2717}{40}\right)\sqrt{2}\right)\left(\left(\dfrac{6029}{168}\right)\sqrt{2}-\dfrac{425}{12}\right)\\
&=&\left(\dfrac{5597441}{2016}\right)\sqrt{2}-\dfrac{13175}{36}+\left(-\dfrac{16380793}{6720}\right)\sqrt{4}\\
&=&\left(\dfrac{5597441}{2016}\right)\sqrt{2}-\dfrac{52831379}{10080}\\
\end{eqnarray*}