L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{62}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)+8+\left(-\dfrac{1}{4}\right)\sqrt{18}+\left(\left(\dfrac{2}{5}\right)\sqrt{8}\right)-0-\left(\left(-\dfrac{79}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)\) et \( Y=\left(\dfrac{3}{2}\right)\sqrt{18}+\left(-\dfrac{16}{3}\right)\sqrt{4}+\left(\dfrac{76}{7}\right)\sqrt{8}+\left(\left(9\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{18}\right)\) . 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{62}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)+8+\left(-\dfrac{1}{4}\right)\sqrt{18}+\left(\left(\dfrac{2}{5}\right)\sqrt{8}\right)-0-\left(\left(-\dfrac{79}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)\right)+\left(\left(\dfrac{3}{2}\right)\sqrt{18}+\left(-\dfrac{16}{3}\right)\sqrt{4}+\left(\dfrac{76}{7}\right)\sqrt{8}+\left(\left(9\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+6+8+\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(\left(\dfrac{4}{5}\right)\sqrt{2}\right)-0-\left(\left(-79\right)\sqrt{2}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)\right)+\left(\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{32}{3}+\left(\dfrac{152}{7}\right)\sqrt{2}+18-\left(\left(-\dfrac{153}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+6+8+\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(\left(\dfrac{4}{5}\right)\sqrt{2}\right)-0-\left(\left(-79\right)\sqrt{2}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)+\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{32}{3}+\left(\dfrac{152}{7}\right)\sqrt{2}+18-\left(\left(-\dfrac{153}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{25033}{210}\right)\sqrt{2}+\dfrac{64}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{62}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)+8+\left(-\dfrac{1}{4}\right)\sqrt{18}+\left(\left(\dfrac{2}{5}\right)\sqrt{8}\right)-0-\left(\left(-\dfrac{79}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{3}{2}\right)\sqrt{18}+\left(-\dfrac{16}{3}\right)\sqrt{4}+\left(\dfrac{76}{7}\right)\sqrt{8}+\left(\left(9\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+6+8+\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(\left(\dfrac{4}{5}\right)\sqrt{2}\right)-0-\left(\left(-79\right)\sqrt{2}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{32}{3}+\left(\dfrac{152}{7}\right)\sqrt{2}+18-\left(\left(-\dfrac{153}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{19841}{420}\right)\sqrt{2}+14\right)-\left(\left(\dfrac{2015}{28}\right)\sqrt{2}+\dfrac{22}{3}\right)\\
&=&\left(\dfrac{19841}{420}\right)\sqrt{2}+14+\left(-\dfrac{2015}{28}\right)\sqrt{2}-\dfrac{22}{3}\\
&=&\left(-\dfrac{2596}{105}\right)\sqrt{2}+\dfrac{20}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{62}{9}\right)\sqrt{18}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{5}\right)\sqrt{50}\right)-\left(\left(-3\right)\sqrt{4}\right)+8+\left(-\dfrac{1}{4}\right)\sqrt{18}+\left(\left(\dfrac{2}{5}\right)\sqrt{8}\right)-0-\left(\left(-\dfrac{79}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{5}\right)\sqrt{50}\right)\right)\times\left(\left(\dfrac{3}{2}\right)\sqrt{18}+\left(-\dfrac{16}{3}\right)\sqrt{4}+\left(\dfrac{76}{7}\right)\sqrt{8}+\left(\left(9\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{41}{4}\right)\sqrt{18}\right)\right)\\
&=&\left(\left(\left(-\dfrac{62}{3}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+6+8+\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(\left(\dfrac{4}{5}\right)\sqrt{2}\right)-0-\left(\left(-79\right)\sqrt{2}\right)-\left(\left(\dfrac{78}{7}\right)\sqrt{2}\right)-\left(\left(-11\right)\sqrt{2}\right)\right)\times\left(\left(\dfrac{9}{2}\right)\sqrt{2}-\dfrac{32}{3}+\left(\dfrac{152}{7}\right)\sqrt{2}+18-\left(\left(-\dfrac{153}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{123}{4}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{19841}{420}\right)\sqrt{2}+14\right)\left(\left(\dfrac{2015}{28}\right)\sqrt{2}+\dfrac{22}{3}\right)\\
&=&\left(\dfrac{7995923}{2352}\right)\sqrt{4}+\left(\dfrac{426488}{315}\right)\sqrt{2}+\dfrac{308}{3}\\
&=&\dfrac{2705553}{392}+\left(\dfrac{426488}{315}\right)\sqrt{2}\\
\end{eqnarray*}