L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\left(\dfrac{78}{7}\right)\sqrt{27}\right)-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{27}+\left(\dfrac{19}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{17}{8}\right)\sqrt{9}\right)+\dfrac{73}{6}\) et \( Y=\left(\left(\dfrac{22}{5}\right)\sqrt{12}+6-\dfrac{5}{9}+\left(\dfrac{22}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{12}+\dfrac{31}{6}+\left(-8\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{12}-8\right)\) . 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{78}{7}\right)\sqrt{27}\right)-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{27}+\left(\dfrac{19}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{17}{8}\right)\sqrt{9}\right)+\dfrac{73}{6}\right)+\left(\left(\left(\dfrac{22}{5}\right)\sqrt{12}+6-\dfrac{5}{9}+\left(\dfrac{22}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{12}+\dfrac{31}{6}+\left(-8\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{12}-8\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{234}{7}\right)\sqrt{3}\right)-15-0-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\dfrac{57}{4}\right)-\dfrac{51}{8}+\dfrac{73}{6}\right)+\left(\left(\left(\dfrac{44}{5}\right)\sqrt{3}+6-\dfrac{5}{9}+\left(\dfrac{44}{5}\right)\sqrt{3}\right)-\left(\left(-19\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{7}\right)\sqrt{3}+\dfrac{31}{6}+\left(-16\right)\sqrt{3}+\left(1\right)\sqrt{3}-8\right)\right)\\
&=&\left(\left(\left(\dfrac{234}{7}\right)\sqrt{3}\right)-15-0-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\dfrac{57}{4}\right)-\dfrac{51}{8}+\dfrac{73}{6}+\left(\left(\dfrac{44}{5}\right)\sqrt{3}+6-\dfrac{5}{9}+\left(\dfrac{44}{5}\right)\sqrt{3}\right)-\left(\left(-19\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{7}\right)\sqrt{3}+\dfrac{31}{6}+\left(-16\right)\sqrt{3}+\left(1\right)\sqrt{3}-8\right)\\
&=&\left(\dfrac{1709}{15}\right)\sqrt{3}-\dfrac{1093}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{78}{7}\right)\sqrt{27}\right)-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{27}+\left(\dfrac{19}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{17}{8}\right)\sqrt{9}\right)+\dfrac{73}{6}\right)-\left(\left(\left(\dfrac{22}{5}\right)\sqrt{12}+6-\dfrac{5}{9}+\left(\dfrac{22}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{12}+\dfrac{31}{6}+\left(-8\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{12}-8\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{234}{7}\right)\sqrt{3}\right)-15-0-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\dfrac{57}{4}\right)-\dfrac{51}{8}+\dfrac{73}{6}\right)-\left(\left(\left(\dfrac{44}{5}\right)\sqrt{3}+6-\dfrac{5}{9}+\left(\dfrac{44}{5}\right)\sqrt{3}\right)-\left(\left(-19\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{7}\right)\sqrt{3}+\dfrac{31}{6}+\left(-16\right)\sqrt{3}+\left(1\right)\sqrt{3}-8\right)\right)\\
&=&\left(\left(\dfrac{835}{21}\right)\sqrt{3}-\dfrac{563}{24}\right)-\left(\left(\dfrac{2596}{35}\right)\sqrt{3}+\dfrac{149}{18}\right)\\
&=&\left(\dfrac{835}{21}\right)\sqrt{3}-\dfrac{563}{24}+\left(-\dfrac{2596}{35}\right)\sqrt{3}-\dfrac{149}{18}\\
&=&\left(-\dfrac{3613}{105}\right)\sqrt{3}-\dfrac{2285}{72}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{78}{7}\right)\sqrt{27}\right)-\left(\left(5\right)\sqrt{9}\right)-\left(\left(0\right)\sqrt{9}\right)-\left(\left(-2\right)\sqrt{75}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{44}{9}\right)\sqrt{27}+\left(\dfrac{19}{4}\right)\sqrt{9}\right)-\left(\left(\dfrac{17}{8}\right)\sqrt{9}\right)+\dfrac{73}{6}\right)\times\left(\left(\left(\dfrac{22}{5}\right)\sqrt{12}+6-\dfrac{5}{9}+\left(\dfrac{22}{5}\right)\sqrt{12}\right)-\left(\left(-\dfrac{19}{2}\right)\sqrt{12}\right)-\left(\left(-\dfrac{79}{7}\right)\sqrt{12}+\dfrac{31}{6}+\left(-8\right)\sqrt{12}+\left(\dfrac{1}{2}\right)\sqrt{12}-8\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{234}{7}\right)\sqrt{3}\right)-15-0-\left(\left(-10\right)\sqrt{3}\right)-\left(\left(\dfrac{55}{3}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{44}{3}\right)\sqrt{3}+\dfrac{57}{4}\right)-\dfrac{51}{8}+\dfrac{73}{6}\right)\times\left(\left(\left(\dfrac{44}{5}\right)\sqrt{3}+6-\dfrac{5}{9}+\left(\dfrac{44}{5}\right)\sqrt{3}\right)-\left(\left(-19\right)\sqrt{3}\right)-\left(\left(-\dfrac{158}{7}\right)\sqrt{3}+\dfrac{31}{6}+\left(-16\right)\sqrt{3}+\left(1\right)\sqrt{3}-8\right)\right)\\
&=&\left(\left(\dfrac{835}{21}\right)\sqrt{3}-\dfrac{563}{24}\right)\left(\left(\dfrac{2596}{35}\right)\sqrt{3}+\dfrac{149}{18}\right)\\
&=&\left(\dfrac{433532}{147}\right)\sqrt{9}+\left(-\dfrac{1333204}{945}\right)\sqrt{3}-\dfrac{83887}{432}\\
&=&\dfrac{183175361}{21168}+\left(-\dfrac{1333204}{945}\right)\sqrt{3}\\
\end{eqnarray*}