L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{23}{3}\right)\sqrt{18}+\left(2\right)\sqrt{50}+\left(\dfrac{15}{2}\right)\sqrt{50}+\left(-\dfrac{50}{3}\right)\sqrt{4}+8\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}+\left(\dfrac{3}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{8}+\left(-3\right)\sqrt{8}+\left(\dfrac{3}{2}\right)\sqrt{50}+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\) et \( Y=\left(8\right)\sqrt{50}+\left(\dfrac{53}{2}\right)\sqrt{4}+\left(-1\right)\sqrt{4}+\left(\dfrac{55}{9}\right)\sqrt{4}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{20}{3}\right)\sqrt{4}+\left(-6\right)\sqrt{50}+\left(\dfrac{24}{7}\right)\sqrt{8}+\left(8\right)\sqrt{50}+\left(9\right)\sqrt{4}\) . 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}{3}\right)\sqrt{18}+\left(2\right)\sqrt{50}+\left(\dfrac{15}{2}\right)\sqrt{50}+\left(-\dfrac{50}{3}\right)\sqrt{4}+8\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}+\left(\dfrac{3}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{8}+\left(-3\right)\sqrt{8}+\left(\dfrac{3}{2}\right)\sqrt{50}+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\right)+\left(\left(8\right)\sqrt{50}+\left(\dfrac{53}{2}\right)\sqrt{4}+\left(-1\right)\sqrt{4}+\left(\dfrac{55}{9}\right)\sqrt{4}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{20}{3}\right)\sqrt{4}+\left(-6\right)\sqrt{50}+\left(\dfrac{24}{7}\right)\sqrt{8}+\left(8\right)\sqrt{50}+\left(9\right)\sqrt{4}\right)\\
&=&\left(\left(\left(23\right)\sqrt{2}+\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}-\dfrac{100}{3}+8\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{4}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{22}{3}\right)\sqrt{2}\right)\right)+\left(\left(40\right)\sqrt{2}+53-2+\dfrac{110}{9}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\dfrac{40}{3}+\left(-30\right)\sqrt{2}+\left(\dfrac{48}{7}\right)\sqrt{2}+\left(40\right)\sqrt{2}+18\right)\\
&=&\left(\left(23\right)\sqrt{2}+\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}-\dfrac{100}{3}+8\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{4}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{22}{3}\right)\sqrt{2}\right)+\left(40\right)\sqrt{2}+53-2+\dfrac{110}{9}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\dfrac{40}{3}+\left(-30\right)\sqrt{2}+\left(\dfrac{48}{7}\right)\sqrt{2}+\left(40\right)\sqrt{2}+18\\
&=&\left(\dfrac{2941}{42}\right)\sqrt{2}+\dfrac{623}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{23}{3}\right)\sqrt{18}+\left(2\right)\sqrt{50}+\left(\dfrac{15}{2}\right)\sqrt{50}+\left(-\dfrac{50}{3}\right)\sqrt{4}+8\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}+\left(\dfrac{3}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{8}+\left(-3\right)\sqrt{8}+\left(\dfrac{3}{2}\right)\sqrt{50}+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\right)-\left(\left(8\right)\sqrt{50}+\left(\dfrac{53}{2}\right)\sqrt{4}+\left(-1\right)\sqrt{4}+\left(\dfrac{55}{9}\right)\sqrt{4}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{20}{3}\right)\sqrt{4}+\left(-6\right)\sqrt{50}+\left(\dfrac{24}{7}\right)\sqrt{8}+\left(8\right)\sqrt{50}+\left(9\right)\sqrt{4}\right)\\
&=&\left(\left(\left(23\right)\sqrt{2}+\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}-\dfrac{100}{3}+8\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{4}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{22}{3}\right)\sqrt{2}\right)\right)-\left(\left(40\right)\sqrt{2}+53-2+\dfrac{110}{9}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\dfrac{40}{3}+\left(-30\right)\sqrt{2}+\left(\dfrac{48}{7}\right)\sqrt{2}+\left(40\right)\sqrt{2}+18\right)\\
&=&\left(\left(\dfrac{227}{3}\right)\sqrt{2}-\dfrac{76}{3}\right)-\left(\left(-\dfrac{79}{14}\right)\sqrt{2}+\dfrac{851}{9}\right)\\
&=&\left(\dfrac{227}{3}\right)\sqrt{2}-\dfrac{76}{3}+\left(\dfrac{79}{14}\right)\sqrt{2}-\dfrac{851}{9}\\
&=&\left(\dfrac{3415}{42}\right)\sqrt{2}-\dfrac{1079}{9}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{23}{3}\right)\sqrt{18}+\left(2\right)\sqrt{50}+\left(\dfrac{15}{2}\right)\sqrt{50}+\left(-\dfrac{50}{3}\right)\sqrt{4}+8\right)-\left(\left(-\dfrac{17}{4}\right)\sqrt{18}+\left(\dfrac{3}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{35}{8}\right)\sqrt{8}+\left(-3\right)\sqrt{8}+\left(\dfrac{3}{2}\right)\sqrt{50}+\left(\dfrac{11}{3}\right)\sqrt{8}\right)\right)\times\left(\left(8\right)\sqrt{50}+\left(\dfrac{53}{2}\right)\sqrt{4}+\left(-1\right)\sqrt{4}+\left(\dfrac{55}{9}\right)\sqrt{4}+\left(-\dfrac{25}{2}\right)\sqrt{50}+\left(\dfrac{20}{3}\right)\sqrt{4}+\left(-6\right)\sqrt{50}+\left(\dfrac{24}{7}\right)\sqrt{8}+\left(8\right)\sqrt{50}+\left(9\right)\sqrt{4}\right)\\
&=&\left(\left(\left(23\right)\sqrt{2}+\left(10\right)\sqrt{2}+\left(\dfrac{75}{2}\right)\sqrt{2}-\dfrac{100}{3}+8\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{4}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+\left(\dfrac{15}{2}\right)\sqrt{2}+\left(\dfrac{22}{3}\right)\sqrt{2}\right)\right)\times\left(\left(40\right)\sqrt{2}+53-2+\dfrac{110}{9}+\left(-\dfrac{125}{2}\right)\sqrt{2}+\dfrac{40}{3}+\left(-30\right)\sqrt{2}+\left(\dfrac{48}{7}\right)\sqrt{2}+\left(40\right)\sqrt{2}+18\right)\\
&=&\left(\left(\dfrac{227}{3}\right)\sqrt{2}-\dfrac{76}{3}\right)\left(\left(-\dfrac{79}{14}\right)\sqrt{2}+\dfrac{851}{9}\right)\\
&=&\left(-\dfrac{17933}{42}\right)\sqrt{4}+\left(\dfrac{1379257}{189}\right)\sqrt{2}-\dfrac{64676}{27}\\
&=&-\dfrac{614129}{189}+\left(\dfrac{1379257}{189}\right)\sqrt{2}\\
\end{eqnarray*}