L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(\dfrac{68}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(4\right)\sqrt{4}+\left(\dfrac{37}{4}\right)\sqrt{50}+\left(-\dfrac{46}{9}\right)\sqrt{50}+\left(-\dfrac{31}{7}\right)\sqrt{8}\right)\) et \( Y=\left(\left(1\right)\sqrt{8}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}+\left(\dfrac{47}{5}\right)\sqrt{8}+\left(\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{13}{2}\right)\sqrt{4}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(\dfrac{68}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(4\right)\sqrt{4}+\left(\dfrac{37}{4}\right)\sqrt{50}+\left(-\dfrac{46}{9}\right)\sqrt{50}+\left(-\dfrac{31}{7}\right)\sqrt{8}\right)\right)+\left(\left(\left(1\right)\sqrt{8}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}+\left(\dfrac{47}{5}\right)\sqrt{8}+\left(\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{13}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{73}{2}-\dfrac{73}{2}+\left(\dfrac{340}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(8+\left(\dfrac{185}{4}\right)\sqrt{2}+\left(-\dfrac{230}{9}\right)\sqrt{2}+\left(-\dfrac{62}{7}\right)\sqrt{2}\right)\right)+\left(\left(\left(2\right)\sqrt{2}\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}+\left(\dfrac{94}{5}\right)\sqrt{2}+\left(\dfrac{80}{7}\right)\sqrt{2}-13\right)\right)\\
&=&\left(-\dfrac{73}{2}-\dfrac{73}{2}+\left(\dfrac{340}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(8+\left(\dfrac{185}{4}\right)\sqrt{2}+\left(-\dfrac{230}{9}\right)\sqrt{2}+\left(-\dfrac{62}{7}\right)\sqrt{2}\right)+\left(\left(2\right)\sqrt{2}\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}+\left(\dfrac{94}{5}\right)\sqrt{2}+\left(\dfrac{80}{7}\right)\sqrt{2}-13\right)\\
&=&-68+\left(\dfrac{130327}{1260}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(\dfrac{68}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(4\right)\sqrt{4}+\left(\dfrac{37}{4}\right)\sqrt{50}+\left(-\dfrac{46}{9}\right)\sqrt{50}+\left(-\dfrac{31}{7}\right)\sqrt{8}\right)\right)-\left(\left(\left(1\right)\sqrt{8}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}+\left(\dfrac{47}{5}\right)\sqrt{8}+\left(\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{13}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{73}{2}-\dfrac{73}{2}+\left(\dfrac{340}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(8+\left(\dfrac{185}{4}\right)\sqrt{2}+\left(-\dfrac{230}{9}\right)\sqrt{2}+\left(-\dfrac{62}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(2\right)\sqrt{2}\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}+\left(\dfrac{94}{5}\right)\sqrt{2}+\left(\dfrac{80}{7}\right)\sqrt{2}-13\right)\right)\\
&=&\left(-81+\left(\dfrac{26963}{252}\right)\sqrt{2}\right)-\left(\left(-\dfrac{374}{105}\right)\sqrt{2}+13\right)\\
&=&-81+\left(\dfrac{26963}{252}\right)\sqrt{2}+\left(\dfrac{374}{105}\right)\sqrt{2}-13\\
&=&-94+\left(\dfrac{139303}{1260}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(-\dfrac{73}{4}\right)\sqrt{4}+\left(\dfrac{68}{3}\right)\sqrt{50}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{8}\right)-\left(\left(4\right)\sqrt{4}+\left(\dfrac{37}{4}\right)\sqrt{50}+\left(-\dfrac{46}{9}\right)\sqrt{50}+\left(-\dfrac{31}{7}\right)\sqrt{8}\right)\right)\times\left(\left(\left(1\right)\sqrt{8}\right)-\left(\left(-\dfrac{37}{3}\right)\sqrt{8}+\left(\dfrac{47}{5}\right)\sqrt{8}+\left(\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{13}{2}\right)\sqrt{4}\right)\right)\\
&=&\left(\left(-\dfrac{73}{2}-\dfrac{73}{2}+\left(\dfrac{340}{3}\right)\sqrt{2}\right)-\left(\left(-\dfrac{11}{2}\right)\sqrt{2}\right)-\left(8+\left(\dfrac{185}{4}\right)\sqrt{2}+\left(-\dfrac{230}{9}\right)\sqrt{2}+\left(-\dfrac{62}{7}\right)\sqrt{2}\right)\right)\times\left(\left(\left(2\right)\sqrt{2}\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}+\left(\dfrac{94}{5}\right)\sqrt{2}+\left(\dfrac{80}{7}\right)\sqrt{2}-13\right)\right)\\
&=&\left(-81+\left(\dfrac{26963}{252}\right)\sqrt{2}\right)\left(\left(-\dfrac{374}{105}\right)\sqrt{2}+13\right)\\
&=&\left(\dfrac{2116123}{1260}\right)\sqrt{2}-1053+\left(-\dfrac{5042081}{13230}\right)\sqrt{4}\\
&=&\left(\dfrac{2116123}{1260}\right)\sqrt{2}-\dfrac{12007676}{6615}\\
\end{eqnarray*}