L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{23}{2}\right)\sqrt{18}+\left(9\right)\sqrt{8}+\left(\dfrac{1}{2}\right)\sqrt{8}+\left(-\dfrac{28}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{50}\right)\right)\) et \( Y=1-\left(\left(\left(\dfrac{67}{9}\right)\sqrt{50}\right)+\dfrac{26}{3}\right)\) . 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(\dfrac{23}{2}\right)\sqrt{18}+\left(9\right)\sqrt{8}+\left(\dfrac{1}{2}\right)\sqrt{8}+\left(-\dfrac{28}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{50}\right)\right)\right)+\left(1-\left(\left(\left(\dfrac{67}{9}\right)\sqrt{50}\right)+\dfrac{26}{3}\right)\right)\\
&=&\left(\left(\left(\dfrac{69}{2}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(1\right)\sqrt{2}+\left(-28\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{2}\right)-\left(\left(78\right)\sqrt{2}\right)\right)\right)+\left(1-\left(\left(\left(\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{26}{3}\right)\right)\\
&=&\left(\left(\dfrac{69}{2}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(1\right)\sqrt{2}+\left(-28\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{2}\right)-\left(\left(78\right)\sqrt{2}\right)\right)+1-\left(\left(\left(\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{26}{3}\right)\\
&=&\left(\dfrac{2539}{36}\right)\sqrt{2}-\dfrac{23}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{23}{2}\right)\sqrt{18}+\left(9\right)\sqrt{8}+\left(\dfrac{1}{2}\right)\sqrt{8}+\left(-\dfrac{28}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{50}\right)\right)\right)-\left(1-\left(\left(\left(\dfrac{67}{9}\right)\sqrt{50}\right)+\dfrac{26}{3}\right)\right)\\
&=&\left(\left(\left(\dfrac{69}{2}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(1\right)\sqrt{2}+\left(-28\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{2}\right)-\left(\left(78\right)\sqrt{2}\right)\right)\right)-\left(1-\left(\left(\left(\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{26}{3}\right)\right)\\
&=&\left(\left(\dfrac{431}{4}\right)\sqrt{2}\right)-\left(-\dfrac{23}{3}+\left(-\dfrac{335}{9}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{431}{4}\right)\sqrt{2}+\dfrac{23}{3}+\left(\dfrac{335}{9}\right)\sqrt{2}\\
&=&\left(\dfrac{5219}{36}\right)\sqrt{2}+\dfrac{23}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{23}{2}\right)\sqrt{18}+\left(9\right)\sqrt{8}+\left(\dfrac{1}{2}\right)\sqrt{8}+\left(-\dfrac{28}{3}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{17}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{78}{5}\right)\sqrt{50}\right)\right)\right)\times\left(1-\left(\left(\left(\dfrac{67}{9}\right)\sqrt{50}\right)+\dfrac{26}{3}\right)\right)\\
&=&\left(\left(\left(\dfrac{69}{2}\right)\sqrt{2}+\left(18\right)\sqrt{2}+\left(1\right)\sqrt{2}+\left(-28\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{17}{4}\right)\sqrt{2}\right)-\left(\left(78\right)\sqrt{2}\right)\right)\right)\times\left(1-\left(\left(\left(\dfrac{335}{9}\right)\sqrt{2}\right)+\dfrac{26}{3}\right)\right)\\
&=&\left(\left(\dfrac{431}{4}\right)\sqrt{2}\right)\left(-\dfrac{23}{3}+\left(-\dfrac{335}{9}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{9913}{12}\right)\sqrt{2}+\left(-\dfrac{144385}{36}\right)\sqrt{4}\\
&=&\left(-\dfrac{9913}{12}\right)\sqrt{2}-\dfrac{144385}{18}\\
\end{eqnarray*}