L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{19}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{18}\right)+\left(\dfrac{16}{7}\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\left(\dfrac{1}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{9}\right)\sqrt{18}\right)-2\) et \( Y=-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)+\left(\left(-8\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)\) . 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{19}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{18}\right)+\left(\dfrac{16}{7}\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\left(\dfrac{1}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{9}\right)\sqrt{18}\right)-2\right)+\left(-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)+\left(\left(-8\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(19\right)\sqrt{2}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\dfrac{48}{7}\right)\sqrt{2}+16+1-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)-2\right)+\left(-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)+\left(\left(-40\right)\sqrt{2}\right)+6\right)\\
&=&\left(\left(19\right)\sqrt{2}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\dfrac{48}{7}\right)\sqrt{2}+16+1-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)-2-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)+\left(\left(-40\right)\sqrt{2}\right)+6\\
&=&\left(-\dfrac{571}{28}\right)\sqrt{2}+\dfrac{108}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{19}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{18}\right)+\left(\dfrac{16}{7}\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\left(\dfrac{1}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{9}\right)\sqrt{18}\right)-2\right)-\left(-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)+\left(\left(-8\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(19\right)\sqrt{2}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\dfrac{48}{7}\right)\sqrt{2}+16+1-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)-2\right)-\left(-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)+\left(\left(-40\right)\sqrt{2}\right)+6\right)\\
&=&\left(\left(\dfrac{69}{7}\right)\sqrt{2}+15\right)-\left(\dfrac{3}{7}+\left(-\dfrac{121}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{69}{7}\right)\sqrt{2}+15+-\dfrac{3}{7}+\left(\dfrac{121}{4}\right)\sqrt{2}\\
&=&\left(\dfrac{1123}{28}\right)\sqrt{2}+\dfrac{102}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{19}{3}\right)\sqrt{18}\right)-\left(\left(-\dfrac{32}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{18}\right)+\left(\dfrac{16}{7}\right)\sqrt{18}+\left(8\right)\sqrt{4}+\left(\left(\dfrac{1}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{37}{9}\right)\sqrt{18}\right)-2\right)\times\left(-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{8}\right)\sqrt{8}\right)+\left(\left(-8\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)\right)\\
&=&\left(\left(\left(19\right)\sqrt{2}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\dfrac{48}{7}\right)\sqrt{2}+16+1-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)-2\right)\times\left(-\dfrac{71}{7}+\dfrac{32}{7}-\left(\left(-\dfrac{39}{4}\right)\sqrt{2}\right)+\left(\left(-40\right)\sqrt{2}\right)+6\right)\\
&=&\left(\left(\dfrac{69}{7}\right)\sqrt{2}+15\right)\left(\dfrac{3}{7}+\left(-\dfrac{121}{4}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{88107}{196}\right)\sqrt{2}+\left(-\dfrac{8349}{28}\right)\sqrt{4}+\dfrac{45}{7}\\
&=&\left(-\dfrac{88107}{196}\right)\sqrt{2}-\dfrac{8259}{14}\\
\end{eqnarray*}