L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(6\right)\sqrt{18}+\left(0\right)\sqrt{8}+\left(-5\right)\sqrt{4}-8+\left(-\dfrac{59}{3}\right)\sqrt{8}-\dfrac{10}{3}+\left(2\right)\sqrt{4}\) et \( Y=\left(3\right)\sqrt{18}+\dfrac{32}{3}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(0\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(6\right)\sqrt{18}+\left(0\right)\sqrt{8}+\left(-5\right)\sqrt{4}-8+\left(-\dfrac{59}{3}\right)\sqrt{8}-\dfrac{10}{3}+\left(2\right)\sqrt{4}\right)+\left(\left(3\right)\sqrt{18}+\dfrac{32}{3}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(0\right)\sqrt{50}\right)\\
&=&\left(\left(18\right)\sqrt{2}+\left(0\right)\sqrt{2}-10-8+\left(-\dfrac{118}{3}\right)\sqrt{2}-\dfrac{10}{3}+4\right)+\left(\left(9\right)\sqrt{2}+\dfrac{32}{3}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(18\right)\sqrt{2}+\left(0\right)\sqrt{2}-10-8+\left(-\dfrac{118}{3}\right)\sqrt{2}-\dfrac{10}{3}+4+\left(9\right)\sqrt{2}+\dfrac{32}{3}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(0\right)\sqrt{2}\\
&=&\left(-\dfrac{1937}{42}\right)\sqrt{2}-\dfrac{20}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(6\right)\sqrt{18}+\left(0\right)\sqrt{8}+\left(-5\right)\sqrt{4}-8+\left(-\dfrac{59}{3}\right)\sqrt{8}-\dfrac{10}{3}+\left(2\right)\sqrt{4}\right)-\left(\left(3\right)\sqrt{18}+\dfrac{32}{3}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(0\right)\sqrt{50}\right)\\
&=&\left(\left(18\right)\sqrt{2}+\left(0\right)\sqrt{2}-10-8+\left(-\dfrac{118}{3}\right)\sqrt{2}-\dfrac{10}{3}+4\right)-\left(\left(9\right)\sqrt{2}+\dfrac{32}{3}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{64}{3}\right)\sqrt{2}-\dfrac{52}{3}\right)-\left(\left(-\dfrac{347}{14}\right)\sqrt{2}+\dfrac{32}{3}\right)\\
&=&\left(-\dfrac{64}{3}\right)\sqrt{2}-\dfrac{52}{3}+\left(\dfrac{347}{14}\right)\sqrt{2}-\dfrac{32}{3}\\
&=&\left(\dfrac{145}{42}\right)\sqrt{2}-28\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(6\right)\sqrt{18}+\left(0\right)\sqrt{8}+\left(-5\right)\sqrt{4}-8+\left(-\dfrac{59}{3}\right)\sqrt{8}-\dfrac{10}{3}+\left(2\right)\sqrt{4}\right)\times\left(\left(3\right)\sqrt{18}+\dfrac{32}{3}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{67}{7}\right)\sqrt{18}+\left(0\right)\sqrt{50}\right)\\
&=&\left(\left(18\right)\sqrt{2}+\left(0\right)\sqrt{2}-10-8+\left(-\dfrac{118}{3}\right)\sqrt{2}-\dfrac{10}{3}+4\right)\times\left(\left(9\right)\sqrt{2}+\dfrac{32}{3}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\left(\dfrac{201}{7}\right)\sqrt{2}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{64}{3}\right)\sqrt{2}-\dfrac{52}{3}\right)\left(\left(-\dfrac{347}{14}\right)\sqrt{2}+\dfrac{32}{3}\right)\\
&=&\left(\dfrac{11104}{21}\right)\sqrt{4}+\left(\dfrac{12730}{63}\right)\sqrt{2}-\dfrac{1664}{9}\\
&=&\dfrac{54976}{63}+\left(\dfrac{12730}{63}\right)\sqrt{2}\\
\end{eqnarray*}