L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(3\right)\sqrt{50}\right)-\left(\left(-\dfrac{16}{9}\right)\sqrt{4}\right)-\dfrac{65}{3}-\left(\left(\dfrac{32}{7}\right)\sqrt{18}+\left(\dfrac{37}{4}\right)\sqrt{18}+\dfrac{1}{5}+\left(\dfrac{3}{8}\right)\sqrt{18}\right)\) et \( Y=\left(-\dfrac{41}{7}\right)\sqrt{8}+\left(\dfrac{23}{6}\right)\sqrt{4}+\left(5\right)\sqrt{50}-\dfrac{43}{7}+\left(-6\right)\sqrt{4}+\left(-\dfrac{46}{5}\right)\sqrt{8}\) . 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{2}{3}\right)\sqrt{50}+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(3\right)\sqrt{50}\right)-\left(\left(-\dfrac{16}{9}\right)\sqrt{4}\right)-\dfrac{65}{3}-\left(\left(\dfrac{32}{7}\right)\sqrt{18}+\left(\dfrac{37}{4}\right)\sqrt{18}+\dfrac{1}{5}+\left(\dfrac{3}{8}\right)\sqrt{18}\right)\right)+\left(\left(-\dfrac{41}{7}\right)\sqrt{8}+\left(\dfrac{23}{6}\right)\sqrt{4}+\left(5\right)\sqrt{50}-\dfrac{43}{7}+\left(-6\right)\sqrt{4}+\left(-\dfrac{46}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{2}-\dfrac{128}{3}+\left(-\dfrac{10}{3}\right)\sqrt{2}+\left(15\right)\sqrt{2}\right)+\dfrac{32}{9}-\dfrac{65}{3}-\left(\left(\dfrac{96}{7}\right)\sqrt{2}+\left(\dfrac{111}{4}\right)\sqrt{2}+\dfrac{1}{5}+\left(\dfrac{9}{8}\right)\sqrt{2}\right)\right)+\left(\left(-\dfrac{82}{7}\right)\sqrt{2}+\dfrac{23}{3}+\left(25\right)\sqrt{2}-\dfrac{43}{7}-12+\left(-\dfrac{92}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{10}{3}\right)\sqrt{2}-\dfrac{128}{3}+\left(-\dfrac{10}{3}\right)\sqrt{2}+\left(15\right)\sqrt{2}\right)+\dfrac{32}{9}-\dfrac{65}{3}-\left(\left(\dfrac{96}{7}\right)\sqrt{2}+\left(\dfrac{111}{4}\right)\sqrt{2}+\dfrac{1}{5}+\left(\dfrac{9}{8}\right)\sqrt{2}\right)+\left(-\dfrac{82}{7}\right)\sqrt{2}+\dfrac{23}{3}+\left(25\right)\sqrt{2}-\dfrac{43}{7}-12+\left(-\dfrac{92}{5}\right)\sqrt{2}\\
&=&\left(-\dfrac{33071}{840}\right)\sqrt{2}-\dfrac{22508}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(3\right)\sqrt{50}\right)-\left(\left(-\dfrac{16}{9}\right)\sqrt{4}\right)-\dfrac{65}{3}-\left(\left(\dfrac{32}{7}\right)\sqrt{18}+\left(\dfrac{37}{4}\right)\sqrt{18}+\dfrac{1}{5}+\left(\dfrac{3}{8}\right)\sqrt{18}\right)\right)-\left(\left(-\dfrac{41}{7}\right)\sqrt{8}+\left(\dfrac{23}{6}\right)\sqrt{4}+\left(5\right)\sqrt{50}-\dfrac{43}{7}+\left(-6\right)\sqrt{4}+\left(-\dfrac{46}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{2}-\dfrac{128}{3}+\left(-\dfrac{10}{3}\right)\sqrt{2}+\left(15\right)\sqrt{2}\right)+\dfrac{32}{9}-\dfrac{65}{3}-\left(\left(\dfrac{96}{7}\right)\sqrt{2}+\left(\dfrac{111}{4}\right)\sqrt{2}+\dfrac{1}{5}+\left(\dfrac{9}{8}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{82}{7}\right)\sqrt{2}+\dfrac{23}{3}+\left(25\right)\sqrt{2}-\dfrac{43}{7}-12+\left(-\dfrac{92}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{5755}{168}\right)\sqrt{2}-\dfrac{2744}{45}\right)-\left(\left(-\dfrac{179}{35}\right)\sqrt{2}-\dfrac{220}{21}\right)\\
&=&\left(-\dfrac{5755}{168}\right)\sqrt{2}-\dfrac{2744}{45}+\left(\dfrac{179}{35}\right)\sqrt{2}+\dfrac{220}{21}\\
&=&\left(-\dfrac{3497}{120}\right)\sqrt{2}-\dfrac{15908}{315}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(-\dfrac{64}{3}\right)\sqrt{4}+\left(-\dfrac{2}{3}\right)\sqrt{50}+\left(3\right)\sqrt{50}\right)-\left(\left(-\dfrac{16}{9}\right)\sqrt{4}\right)-\dfrac{65}{3}-\left(\left(\dfrac{32}{7}\right)\sqrt{18}+\left(\dfrac{37}{4}\right)\sqrt{18}+\dfrac{1}{5}+\left(\dfrac{3}{8}\right)\sqrt{18}\right)\right)\times\left(\left(-\dfrac{41}{7}\right)\sqrt{8}+\left(\dfrac{23}{6}\right)\sqrt{4}+\left(5\right)\sqrt{50}-\dfrac{43}{7}+\left(-6\right)\sqrt{4}+\left(-\dfrac{46}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(-\dfrac{10}{3}\right)\sqrt{2}-\dfrac{128}{3}+\left(-\dfrac{10}{3}\right)\sqrt{2}+\left(15\right)\sqrt{2}\right)+\dfrac{32}{9}-\dfrac{65}{3}-\left(\left(\dfrac{96}{7}\right)\sqrt{2}+\left(\dfrac{111}{4}\right)\sqrt{2}+\dfrac{1}{5}+\left(\dfrac{9}{8}\right)\sqrt{2}\right)\right)\times\left(\left(-\dfrac{82}{7}\right)\sqrt{2}+\dfrac{23}{3}+\left(25\right)\sqrt{2}-\dfrac{43}{7}-12+\left(-\dfrac{92}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{5755}{168}\right)\sqrt{2}-\dfrac{2744}{45}\right)\left(\left(-\dfrac{179}{35}\right)\sqrt{2}-\dfrac{220}{21}\right)\\
&=&\left(\dfrac{206029}{1176}\right)\sqrt{4}+\left(\dfrac{4929863}{7350}\right)\sqrt{2}+\dfrac{17248}{27}\\
&=&\dfrac{5234869}{5292}+\left(\dfrac{4929863}{7350}\right)\sqrt{2}\\
\end{eqnarray*}