L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(-\dfrac{24}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-1\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{9}+\left(\dfrac{71}{2}\right)\sqrt{12}+\left(-\dfrac{28}{9}\right)\sqrt{27}\right)\) et \( Y=\left(\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(-6\right)\sqrt{12}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{75}\right)\) . 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(\left(-\dfrac{24}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-1\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{9}+\left(\dfrac{71}{2}\right)\sqrt{12}+\left(-\dfrac{28}{9}\right)\sqrt{27}\right)\right)+\left(\left(\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(-6\right)\sqrt{12}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{72}{5}-\dfrac{129}{4}-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-3+3\right)-\left(-23+\left(71\right)\sqrt{3}+\left(-\dfrac{28}{3}\right)\sqrt{3}\right)\right)+\left(\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-27\right)\sqrt{3}\right)+55\right)-\dfrac{45}{2}-\left(\left(\left(-\dfrac{135}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{220}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{72}{5}-\dfrac{129}{4}-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-3+3\right)-\left(-23+\left(71\right)\sqrt{3}+\left(-\dfrac{28}{3}\right)\sqrt{3}\right)+\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-27\right)\sqrt{3}\right)+55\right)-\dfrac{45}{2}-\left(\left(\left(-\dfrac{135}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{220}{9}\right)\sqrt{3}\right)\\
&=&\dfrac{177}{20}+\left(-\dfrac{1087}{126}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{24}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-1\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{9}+\left(\dfrac{71}{2}\right)\sqrt{12}+\left(-\dfrac{28}{9}\right)\sqrt{27}\right)\right)-\left(\left(\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(-6\right)\sqrt{12}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{72}{5}-\dfrac{129}{4}-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-3+3\right)-\left(-23+\left(71\right)\sqrt{3}+\left(-\dfrac{28}{3}\right)\sqrt{3}\right)\right)-\left(\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-27\right)\sqrt{3}\right)+55\right)-\dfrac{45}{2}-\left(\left(\left(-\dfrac{135}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{220}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{473}{20}+\left(-\dfrac{2105}{21}\right)\sqrt{3}\right)-\left(\left(\dfrac{1649}{18}\right)\sqrt{3}+\dfrac{65}{2}\right)\\
&=&-\dfrac{473}{20}+\left(-\dfrac{2105}{21}\right)\sqrt{3}+\left(-\dfrac{1649}{18}\right)\sqrt{3}-\dfrac{65}{2}\\
&=&-\dfrac{1123}{20}+\left(-\dfrac{24173}{126}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{24}{5}\right)\sqrt{9}\right)-\left(\left(\dfrac{43}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{54}{7}\right)\sqrt{75}\right)-\left(\left(1\right)\sqrt{9}\right)-\left(\left(-1\right)\sqrt{9}\right)\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{9}+\left(\dfrac{71}{2}\right)\sqrt{12}+\left(-\dfrac{28}{9}\right)\sqrt{27}\right)\right)\times\left(\left(\left(\left(\dfrac{5}{2}\right)\sqrt{12}\right)-\left(\left(-9\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{9}\right)\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{9}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{61}{6}\right)\sqrt{12}\right)-\left(\left(-6\right)\sqrt{12}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(-\dfrac{72}{5}-\dfrac{129}{4}-\left(\left(\dfrac{270}{7}\right)\sqrt{3}\right)-3+3\right)-\left(-23+\left(71\right)\sqrt{3}+\left(-\dfrac{28}{3}\right)\sqrt{3}\right)\right)\times\left(\left(\left(\left(5\right)\sqrt{3}\right)-\left(\left(-27\right)\sqrt{3}\right)+55\right)-\dfrac{45}{2}-\left(\left(\left(-\dfrac{135}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{3}\right)-\left(\left(-12\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{220}{9}\right)\sqrt{3}\right)\right)\\
&=&\left(-\dfrac{473}{20}+\left(-\dfrac{2105}{21}\right)\sqrt{3}\right)\left(\left(\dfrac{1649}{18}\right)\sqrt{3}+\dfrac{65}{2}\right)\\
&=&\left(-\dfrac{13669339}{2520}\right)\sqrt{3}-\dfrac{6149}{8}+\left(-\dfrac{3471145}{378}\right)\sqrt{9}\\
&=&\left(-\dfrac{13669339}{2520}\right)\sqrt{3}-\dfrac{14271967}{504}\\
\end{eqnarray*}