L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\dfrac{15}{2}\) et \( Y=\left(\left(\dfrac{15}{2}\right)\sqrt{4}+\left(-\dfrac{76}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}+\left(-\dfrac{51}{7}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{50}+\left(8\right)\sqrt{8}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{15}{2}\right)+\left(\left(\left(\dfrac{15}{2}\right)\sqrt{4}+\left(-\dfrac{76}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}+\left(-\dfrac{51}{7}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{50}+\left(8\right)\sqrt{8}\right)\right)\\
&=&\left(\dfrac{15}{2}\right)+\left(\left(15+\left(-\dfrac{228}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{275}{3}\right)\sqrt{2}\right)-\left(\dfrac{35}{2}-\dfrac{102}{7}+\left(-\dfrac{320}{7}\right)\sqrt{2}+\left(16\right)\sqrt{2}\right)\right)\\
&=&\dfrac{15}{2}+\left(15+\left(-\dfrac{228}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{275}{3}\right)\sqrt{2}\right)-\left(\dfrac{35}{2}-\dfrac{102}{7}+\left(-\dfrac{320}{7}\right)\sqrt{2}+\left(16\right)\sqrt{2}\right)\\
&=&\dfrac{137}{7}+\left(\dfrac{7957}{105}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{15}{2}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{4}+\left(-\dfrac{76}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}+\left(-\dfrac{51}{7}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{50}+\left(8\right)\sqrt{8}\right)\right)\\
&=&\left(\dfrac{15}{2}\right)-\left(\left(15+\left(-\dfrac{228}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{275}{3}\right)\sqrt{2}\right)-\left(\dfrac{35}{2}-\dfrac{102}{7}+\left(-\dfrac{320}{7}\right)\sqrt{2}+\left(16\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{15}{2}\right)-\left(\dfrac{169}{14}+\left(\dfrac{7957}{105}\right)\sqrt{2}\right)\\
&=&\dfrac{15}{2}+-\dfrac{169}{14}+\left(-\dfrac{7957}{105}\right)\sqrt{2}\\
&=&-\dfrac{32}{7}+\left(-\dfrac{7957}{105}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{15}{2}\right)\times\left(\left(\left(\dfrac{15}{2}\right)\sqrt{4}+\left(-\dfrac{76}{5}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{35}{4}\right)\sqrt{4}+\left(-\dfrac{51}{7}\right)\sqrt{4}+\left(-\dfrac{64}{7}\right)\sqrt{50}+\left(8\right)\sqrt{8}\right)\right)\\
&=&\left(\dfrac{15}{2}\right)\times\left(\left(15+\left(-\dfrac{228}{5}\right)\sqrt{2}\right)-\left(\left(-\dfrac{275}{3}\right)\sqrt{2}\right)-\left(\dfrac{35}{2}-\dfrac{102}{7}+\left(-\dfrac{320}{7}\right)\sqrt{2}+\left(16\right)\sqrt{2}\right)\right)\\
&=&\left(\dfrac{15}{2}\right)\left(\dfrac{169}{14}+\left(\dfrac{7957}{105}\right)\sqrt{2}\right)\\
&=&\dfrac{2535}{28}+\left(\dfrac{7957}{14}\right)\sqrt{2}\\
&=&\dfrac{2535}{28}+\left(\dfrac{7957}{14}\right)\sqrt{2}\\
\end{eqnarray*}