L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{23}{2}\right)\sqrt{8}\) et \( Y=\left(\dfrac{11}{3}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-\dfrac{59}{2}\right)\sqrt{50}-\dfrac{14}{3}+\left(-4\right)\sqrt{18}+\left(-\dfrac{68}{7}\right)\sqrt{4}+\left(\dfrac{33}{2}\right)\sqrt{50}+\left(\dfrac{57}{2}\right)\sqrt{8}\) . 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{23}{2}\right)\sqrt{8}\right)+\left(\left(\dfrac{11}{3}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-\dfrac{59}{2}\right)\sqrt{50}-\dfrac{14}{3}+\left(-4\right)\sqrt{18}+\left(-\dfrac{68}{7}\right)\sqrt{4}+\left(\dfrac{33}{2}\right)\sqrt{50}+\left(\dfrac{57}{2}\right)\sqrt{8}\right)\\
&=&\left(\left(-23\right)\sqrt{2}\right)+\left(\left(\dfrac{22}{3}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-\dfrac{295}{2}\right)\sqrt{2}-\dfrac{14}{3}+\left(-12\right)\sqrt{2}-\dfrac{136}{7}+\left(\dfrac{165}{2}\right)\sqrt{2}+\left(57\right)\sqrt{2}\right)\\
&=&\left(-23\right)\sqrt{2}+\left(\dfrac{22}{3}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-\dfrac{295}{2}\right)\sqrt{2}-\dfrac{14}{3}+\left(-12\right)\sqrt{2}-\dfrac{136}{7}+\left(\dfrac{165}{2}\right)\sqrt{2}+\left(57\right)\sqrt{2}\\
&=&\left(-\dfrac{143}{3}\right)\sqrt{2}-\dfrac{506}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{23}{2}\right)\sqrt{8}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-\dfrac{59}{2}\right)\sqrt{50}-\dfrac{14}{3}+\left(-4\right)\sqrt{18}+\left(-\dfrac{68}{7}\right)\sqrt{4}+\left(\dfrac{33}{2}\right)\sqrt{50}+\left(\dfrac{57}{2}\right)\sqrt{8}\right)\\
&=&\left(\left(-23\right)\sqrt{2}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-\dfrac{295}{2}\right)\sqrt{2}-\dfrac{14}{3}+\left(-12\right)\sqrt{2}-\dfrac{136}{7}+\left(\dfrac{165}{2}\right)\sqrt{2}+\left(57\right)\sqrt{2}\right)\\
&=&\left(\left(-23\right)\sqrt{2}\right)-\left(\left(-\dfrac{74}{3}\right)\sqrt{2}-\dfrac{506}{21}\right)\\
&=&\left(-23\right)\sqrt{2}+\left(\dfrac{74}{3}\right)\sqrt{2}+\dfrac{506}{21}\\
&=&\left(\dfrac{5}{3}\right)\sqrt{2}+\dfrac{506}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{23}{2}\right)\sqrt{8}\right)\times\left(\left(\dfrac{11}{3}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-\dfrac{59}{2}\right)\sqrt{50}-\dfrac{14}{3}+\left(-4\right)\sqrt{18}+\left(-\dfrac{68}{7}\right)\sqrt{4}+\left(\dfrac{33}{2}\right)\sqrt{50}+\left(\dfrac{57}{2}\right)\sqrt{8}\right)\\
&=&\left(\left(-23\right)\sqrt{2}\right)\times\left(\left(\dfrac{22}{3}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-\dfrac{295}{2}\right)\sqrt{2}-\dfrac{14}{3}+\left(-12\right)\sqrt{2}-\dfrac{136}{7}+\left(\dfrac{165}{2}\right)\sqrt{2}+\left(57\right)\sqrt{2}\right)\\
&=&\left(\left(-23\right)\sqrt{2}\right)\left(\left(-\dfrac{74}{3}\right)\sqrt{2}-\dfrac{506}{21}\right)\\
&=&\left(\dfrac{1702}{3}\right)\sqrt{4}+\left(\dfrac{11638}{21}\right)\sqrt{2}\\
&=&\dfrac{3404}{3}+\left(\dfrac{11638}{21}\right)\sqrt{2}\\
\end{eqnarray*}