L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-5\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{4}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(-8\right)\sqrt{8}+\left(-\dfrac{49}{8}\right)\sqrt{50}\right)-\left(\left(6\right)\sqrt{4}\right)\) et \( Y=\left(\dfrac{17}{3}\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-5\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{4}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(-8\right)\sqrt{8}+\left(-\dfrac{49}{8}\right)\sqrt{50}\right)-\left(\left(6\right)\sqrt{4}\right)\right)+\left(\left(\dfrac{17}{3}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{11}{4}\right)-\left(\left(-15\right)\sqrt{2}-1+\left(-16\right)\sqrt{2}+\left(-\dfrac{245}{8}\right)\sqrt{2}\right)-12\right)+\left(\left(\dfrac{85}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{11}{4}\right)-\left(\left(-15\right)\sqrt{2}-1+\left(-16\right)\sqrt{2}+\left(-\dfrac{245}{8}\right)\sqrt{2}\right)-12+\left(\dfrac{85}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{2153}{24}\right)\sqrt{2}-\dfrac{33}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-5\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{4}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(-8\right)\sqrt{8}+\left(-\dfrac{49}{8}\right)\sqrt{50}\right)-\left(\left(6\right)\sqrt{4}\right)\right)-\left(\left(\dfrac{17}{3}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{11}{4}\right)-\left(\left(-15\right)\sqrt{2}-1+\left(-16\right)\sqrt{2}+\left(-\dfrac{245}{8}\right)\sqrt{2}\right)-12\right)-\left(\left(\dfrac{85}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{491}{8}\right)\sqrt{2}-\dfrac{33}{4}\right)-\left(\left(\dfrac{85}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{491}{8}\right)\sqrt{2}-\dfrac{33}{4}+\left(-\dfrac{85}{3}\right)\sqrt{2}\\
&=&\left(\dfrac{793}{24}\right)\sqrt{2}-\dfrac{33}{4}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-5\right)\sqrt{8}\right)-\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{11}{8}\right)\sqrt{4}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(-\dfrac{1}{2}\right)\sqrt{4}+\left(-8\right)\sqrt{8}+\left(-\dfrac{49}{8}\right)\sqrt{50}\right)-\left(\left(6\right)\sqrt{4}\right)\right)\times\left(\left(\dfrac{17}{3}\right)\sqrt{50}\right)\\
&=&\left(\left(\left(-10\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{11}{4}\right)-\left(\left(-15\right)\sqrt{2}-1+\left(-16\right)\sqrt{2}+\left(-\dfrac{245}{8}\right)\sqrt{2}\right)-12\right)\times\left(\left(\dfrac{85}{3}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{491}{8}\right)\sqrt{2}-\dfrac{33}{4}\right)\left(\left(\dfrac{85}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{41735}{24}\right)\sqrt{4}+\left(-\dfrac{935}{4}\right)\sqrt{2}\\
&=&\dfrac{41735}{12}+\left(-\dfrac{935}{4}\right)\sqrt{2}\\
\end{eqnarray*}