L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{15}{2}\right)\sqrt{18}\) et \( Y=\left(2\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(-\dfrac{74}{7}\right)\sqrt{8}+\left(\dfrac{59}{6}\right)\sqrt{8}+\left(\dfrac{27}{8}\right)\sqrt{8}+\dfrac{8}{9}+\left(\dfrac{4}{3}\right)\sqrt{8}\) . 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{15}{2}\right)\sqrt{18}\right)+\left(\left(2\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(-\dfrac{74}{7}\right)\sqrt{8}+\left(\dfrac{59}{6}\right)\sqrt{8}+\left(\dfrac{27}{8}\right)\sqrt{8}+\dfrac{8}{9}+\left(\dfrac{4}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)+\left(\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(-\dfrac{148}{7}\right)\sqrt{2}+\left(\dfrac{59}{3}\right)\sqrt{2}+\left(\dfrac{27}{4}\right)\sqrt{2}+\dfrac{8}{9}+\left(\dfrac{8}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{45}{2}\right)\sqrt{2}+\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(-\dfrac{148}{7}\right)\sqrt{2}+\left(\dfrac{59}{3}\right)\sqrt{2}+\left(\dfrac{27}{4}\right)\sqrt{2}+\dfrac{8}{9}+\left(\dfrac{8}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{2767}{84}\right)\sqrt{2}+\dfrac{8}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{15}{2}\right)\sqrt{18}\right)-\left(\left(2\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(-\dfrac{74}{7}\right)\sqrt{8}+\left(\dfrac{59}{6}\right)\sqrt{8}+\left(\dfrac{27}{8}\right)\sqrt{8}+\dfrac{8}{9}+\left(\dfrac{4}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)-\left(\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(-\dfrac{148}{7}\right)\sqrt{2}+\left(\dfrac{59}{3}\right)\sqrt{2}+\left(\dfrac{27}{4}\right)\sqrt{2}+\dfrac{8}{9}+\left(\dfrac{8}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{4657}{84}\right)\sqrt{2}+\dfrac{8}{9}\right)\\
&=&\left(-\dfrac{45}{2}\right)\sqrt{2}+\left(-\dfrac{4657}{84}\right)\sqrt{2}-\dfrac{8}{9}\\
&=&\left(-\dfrac{6547}{84}\right)\sqrt{2}-\dfrac{8}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{15}{2}\right)\sqrt{18}\right)\times\left(\left(2\right)\sqrt{50}+\left(\dfrac{25}{2}\right)\sqrt{18}+\left(-\dfrac{74}{7}\right)\sqrt{8}+\left(\dfrac{59}{6}\right)\sqrt{8}+\left(\dfrac{27}{8}\right)\sqrt{8}+\dfrac{8}{9}+\left(\dfrac{4}{3}\right)\sqrt{8}\right)\\
&=&\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)\times\left(\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}+\left(-\dfrac{148}{7}\right)\sqrt{2}+\left(\dfrac{59}{3}\right)\sqrt{2}+\left(\dfrac{27}{4}\right)\sqrt{2}+\dfrac{8}{9}+\left(\dfrac{8}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{45}{2}\right)\sqrt{2}\right)\left(\left(\dfrac{4657}{84}\right)\sqrt{2}+\dfrac{8}{9}\right)\\
&=&\left(-\dfrac{69855}{56}\right)\sqrt{4}+\left(-20\right)\sqrt{2}\\
&=&-\dfrac{69855}{28}+\left(-20\right)\sqrt{2}\\
\end{eqnarray*}