L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(-\dfrac{29}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{27}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\) et \( Y=\left(\left(-7\right)\sqrt{27}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{45}{8}\right)\sqrt{12}+\left(4\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{75}+\left(9\right)\sqrt{12}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{55}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{75}\right)\) . 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{29}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{27}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\right)+\left(\left(\left(-7\right)\sqrt{27}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{45}{8}\right)\sqrt{12}+\left(4\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{75}+\left(9\right)\sqrt{12}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{55}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{87}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{165}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{2}\right)\sqrt{3}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\right)+\left(\left(\left(-21\right)\sqrt{3}+\left(\dfrac{108}{5}\right)\sqrt{3}+\left(-\dfrac{45}{4}\right)\sqrt{3}+\left(20\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}+\left(18\right)\sqrt{3}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{275}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\left(-\dfrac{87}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{165}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{2}\right)\sqrt{3}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)+\left(\left(-21\right)\sqrt{3}+\left(\dfrac{108}{5}\right)\sqrt{3}+\left(-\dfrac{45}{4}\right)\sqrt{3}+\left(20\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}+\left(18\right)\sqrt{3}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{275}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)\\
&=&\left(\dfrac{73483}{840}\right)\sqrt{3}+\dfrac{2663}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{29}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{27}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\right)-\left(\left(\left(-7\right)\sqrt{27}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{45}{8}\right)\sqrt{12}+\left(4\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{75}+\left(9\right)\sqrt{12}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{55}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{87}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{165}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{2}\right)\sqrt{3}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\right)-\left(\left(\left(-21\right)\sqrt{3}+\left(\dfrac{108}{5}\right)\sqrt{3}+\left(-\dfrac{45}{4}\right)\sqrt{3}+\left(20\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}+\left(18\right)\sqrt{3}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{275}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{618}{5}\right)\sqrt{3}+59\right)-\left(\left(-\dfrac{30341}{840}\right)\sqrt{3}+\dfrac{185}{42}\right)\\
&=&\left(\dfrac{618}{5}\right)\sqrt{3}+59+\left(\dfrac{30341}{840}\right)\sqrt{3}-\dfrac{185}{42}\\
&=&\left(\dfrac{26833}{168}\right)\sqrt{3}+\dfrac{2293}{42}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{29}{5}\right)\sqrt{27}\right)-\left(\left(-\dfrac{55}{2}\right)\sqrt{27}\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{27}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\right)\times\left(\left(\left(-7\right)\sqrt{27}+\left(\dfrac{36}{5}\right)\sqrt{27}+\left(-\dfrac{45}{8}\right)\sqrt{12}+\left(4\right)\sqrt{75}\right)-\left(\left(\dfrac{2}{7}\right)\sqrt{75}+\left(9\right)\sqrt{12}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{55}{8}\right)\sqrt{75}\right)-\left(\left(-\dfrac{5}{3}\right)\sqrt{75}\right)\right)\\
&=&\left(\left(\left(\left(-\dfrac{87}{5}\right)\sqrt{3}\right)-\left(\left(-\dfrac{165}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{117}{2}\right)\sqrt{3}\right)+\dfrac{52}{3}\right)-\left(-\dfrac{52}{3}-\dfrac{73}{3}\right)\right)\times\left(\left(\left(-21\right)\sqrt{3}+\left(\dfrac{108}{5}\right)\sqrt{3}+\left(-\dfrac{45}{4}\right)\sqrt{3}+\left(20\right)\sqrt{3}\right)-\left(\left(\dfrac{10}{7}\right)\sqrt{3}+\left(18\right)\sqrt{3}+\dfrac{43}{6}-\dfrac{81}{7}+\left(\dfrac{275}{8}\right)\sqrt{3}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{618}{5}\right)\sqrt{3}+59\right)\left(\left(-\dfrac{30341}{840}\right)\sqrt{3}+\dfrac{185}{42}\right)\\
&=&\left(-\dfrac{3125123}{700}\right)\sqrt{9}+\left(-\dfrac{1332799}{840}\right)\sqrt{3}+\dfrac{10915}{42}\\
&=&-\dfrac{3940051}{300}+\left(-\dfrac{1332799}{840}\right)\sqrt{3}\\
\end{eqnarray*}