L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{70}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{18}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{6}{7}\right)\sqrt{50}\right)\right)-\left(\left(\left(2\right)\sqrt{18}\right)-8-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(-1\right)\sqrt{50}\right)\) et \( Y=\dfrac{71}{3}+\left(0\right)\sqrt{18}-\dfrac{1}{3}+\left(0\right)\sqrt{50}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(\left(\dfrac{70}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{18}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{6}{7}\right)\sqrt{50}\right)\right)-\left(\left(\left(2\right)\sqrt{18}\right)-8-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(-1\right)\sqrt{50}\right)\right)+\left(\dfrac{71}{3}+\left(0\right)\sqrt{18}-\dfrac{1}{3}+\left(0\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{140}{9}-\left(\left(-\dfrac{69}{7}\right)\sqrt{2}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{30}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(6\right)\sqrt{2}\right)-8-18\right)-\left(\left(-5\right)\sqrt{2}\right)\right)+\left(\dfrac{71}{3}+\left(0\right)\sqrt{2}-\dfrac{1}{3}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\dfrac{140}{9}-\left(\left(-\dfrac{69}{7}\right)\sqrt{2}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{30}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(6\right)\sqrt{2}\right)-8-18\right)-\left(\left(-5\right)\sqrt{2}\right)+\dfrac{71}{3}+\left(0\right)\sqrt{2}-\dfrac{1}{3}+\left(0\right)\sqrt{2}\\
&=&\dfrac{3485}{63}+\left(\dfrac{92}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{70}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{18}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{6}{7}\right)\sqrt{50}\right)\right)-\left(\left(\left(2\right)\sqrt{18}\right)-8-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(-1\right)\sqrt{50}\right)\right)-\left(\dfrac{71}{3}+\left(0\right)\sqrt{18}-\dfrac{1}{3}+\left(0\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{140}{9}-\left(\left(-\dfrac{69}{7}\right)\sqrt{2}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{30}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(6\right)\sqrt{2}\right)-8-18\right)-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\dfrac{71}{3}+\left(0\right)\sqrt{2}-\dfrac{1}{3}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2015}{63}+\left(\dfrac{92}{7}\right)\sqrt{2}\right)-\left(\dfrac{70}{3}+\left(0\right)\sqrt{2}\right)\\
&=&\dfrac{2015}{63}+\left(\dfrac{92}{7}\right)\sqrt{2}+-\dfrac{70}{3}+\left(0\right)\sqrt{2}\\
&=&\dfrac{545}{63}+\left(\dfrac{92}{7}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{70}{9}\right)\sqrt{4}\right)-\left(\left(-\dfrac{23}{7}\right)\sqrt{18}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{6}{7}\right)\sqrt{50}\right)\right)-\left(\left(\left(2\right)\sqrt{18}\right)-8-\left(\left(9\right)\sqrt{4}\right)\right)-\left(\left(-1\right)\sqrt{50}\right)\right)\times\left(\dfrac{71}{3}+\left(0\right)\sqrt{18}-\dfrac{1}{3}+\left(0\right)\sqrt{50}\right)\\
&=&\left(\left(\dfrac{140}{9}-\left(\left(-\dfrac{69}{7}\right)\sqrt{2}\right)\right)-\left(\dfrac{67}{7}-\left(\left(\dfrac{30}{7}\right)\sqrt{2}\right)\right)-\left(\left(\left(6\right)\sqrt{2}\right)-8-18\right)-\left(\left(-5\right)\sqrt{2}\right)\right)\times\left(\dfrac{71}{3}+\left(0\right)\sqrt{2}-\dfrac{1}{3}+\left(0\right)\sqrt{2}\right)\\
&=&\left(\dfrac{2015}{63}+\left(\dfrac{92}{7}\right)\sqrt{2}\right)\left(\dfrac{70}{3}+\left(0\right)\sqrt{2}\right)\\
&=&\dfrac{20150}{27}+\left(\dfrac{920}{3}\right)\sqrt{2}+\left(0\right)\sqrt{4}\\
&=&\dfrac{20150}{27}+\left(\dfrac{920}{3}\right)\sqrt{2}\\
\end{eqnarray*}