L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{27}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-1\right)\sqrt{75}+\left(-\dfrac{26}{9}\right)\sqrt{75}\right)\) et \( Y=\left(-\dfrac{4}{3}\right)\sqrt{9}+\left(4\right)\sqrt{75}+\left(-\dfrac{3}{4}\right)\sqrt{12}+\left(-\dfrac{41}{5}\right)\sqrt{75}+\dfrac{77}{5}+\left(-3\right)\sqrt{12}+\left(6\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{27}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-1\right)\sqrt{75}+\left(-\dfrac{26}{9}\right)\sqrt{75}\right)\right)+\left(\left(-\dfrac{4}{3}\right)\sqrt{9}+\left(4\right)\sqrt{75}+\left(-\dfrac{3}{4}\right)\sqrt{12}+\left(-\dfrac{41}{5}\right)\sqrt{75}+\dfrac{77}{5}+\left(-3\right)\sqrt{12}+\left(6\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{3}\right)-\left(\left(9\right)\sqrt{3}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-5\right)\sqrt{3}+\left(-\dfrac{130}{9}\right)\sqrt{3}\right)\right)+\left(-4+\left(20\right)\sqrt{3}+\left(-\dfrac{3}{2}\right)\sqrt{3}+\left(-41\right)\sqrt{3}+\dfrac{77}{5}+\left(-6\right)\sqrt{3}+18\right)\\
&=&\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{3}\right)-\left(\left(9\right)\sqrt{3}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-5\right)\sqrt{3}+\left(-\dfrac{130}{9}\right)\sqrt{3}\right)-4+\left(20\right)\sqrt{3}+\left(-\dfrac{3}{2}\right)\sqrt{3}+\left(-41\right)\sqrt{3}+\dfrac{77}{5}+\left(-6\right)\sqrt{3}+18\\
&=&\dfrac{7491}{140}+\left(-\dfrac{1399}{18}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{27}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-1\right)\sqrt{75}+\left(-\dfrac{26}{9}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{9}+\left(4\right)\sqrt{75}+\left(-\dfrac{3}{4}\right)\sqrt{12}+\left(-\dfrac{41}{5}\right)\sqrt{75}+\dfrac{77}{5}+\left(-3\right)\sqrt{12}+\left(6\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{3}\right)-\left(\left(9\right)\sqrt{3}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-5\right)\sqrt{3}+\left(-\dfrac{130}{9}\right)\sqrt{3}\right)\right)-\left(-4+\left(20\right)\sqrt{3}+\left(-\dfrac{3}{2}\right)\sqrt{3}+\left(-41\right)\sqrt{3}+\dfrac{77}{5}+\left(-6\right)\sqrt{3}+18\right)\\
&=&\left(\dfrac{675}{28}+\left(-\dfrac{443}{9}\right)\sqrt{3}\right)-\left(\dfrac{147}{5}+\left(-\dfrac{57}{2}\right)\sqrt{3}\right)\\
&=&\dfrac{675}{28}+\left(-\dfrac{443}{9}\right)\sqrt{3}+-\dfrac{147}{5}+\left(\dfrac{57}{2}\right)\sqrt{3}\\
&=&-\dfrac{741}{140}+\left(-\dfrac{373}{18}\right)\sqrt{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{4}\right)\sqrt{12}\right)-\left(\left(\dfrac{25}{3}\right)\sqrt{12}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{27}\right)-\left(\left(3\right)\sqrt{27}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-1\right)\sqrt{75}+\left(-\dfrac{26}{9}\right)\sqrt{75}\right)\right)\times\left(\left(-\dfrac{4}{3}\right)\sqrt{9}+\left(4\right)\sqrt{75}+\left(-\dfrac{3}{4}\right)\sqrt{12}+\left(-\dfrac{41}{5}\right)\sqrt{75}+\dfrac{77}{5}+\left(-3\right)\sqrt{12}+\left(6\right)\sqrt{9}\right)\\
&=&\left(\left(\dfrac{77}{4}-\left(\left(\dfrac{71}{2}\right)\sqrt{3}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{3}\right)-\left(\left(\dfrac{15}{2}\right)\sqrt{3}\right)-\left(\left(9\right)\sqrt{3}\right)\right)-\left(-\dfrac{45}{7}+8-\dfrac{45}{7}+\left(-5\right)\sqrt{3}+\left(-\dfrac{130}{9}\right)\sqrt{3}\right)\right)\times\left(-4+\left(20\right)\sqrt{3}+\left(-\dfrac{3}{2}\right)\sqrt{3}+\left(-41\right)\sqrt{3}+\dfrac{77}{5}+\left(-6\right)\sqrt{3}+18\right)\\
&=&\left(\dfrac{675}{28}+\left(-\dfrac{443}{9}\right)\sqrt{3}\right)\left(\dfrac{147}{5}+\left(-\dfrac{57}{2}\right)\sqrt{3}\right)\\
&=&\dfrac{2835}{4}+\left(-\dfrac{1792717}{840}\right)\sqrt{3}+\left(\dfrac{8417}{6}\right)\sqrt{9}\\
&=&\dfrac{19669}{4}+\left(-\dfrac{1792717}{840}\right)\sqrt{3}\\
\end{eqnarray*}