L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{31}{6}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{75}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{74}{3}\right)\sqrt{75}-\dfrac{43}{8}\right)\) et \( Y=\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{27}\right)-\dfrac{23}{3}-\left(\left(-3\right)\sqrt{75}\right)\right)-\left(\left(\left(\dfrac{38}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\) . 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(\dfrac{31}{6}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{75}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{74}{3}\right)\sqrt{75}-\dfrac{43}{8}\right)\right)+\left(\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{27}\right)-\dfrac{23}{3}-\left(\left(-3\right)\sqrt{75}\right)\right)-\left(\left(\left(\dfrac{38}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(\dfrac{155}{6}\right)\sqrt{3}\right)-\left(\left(\dfrac{275}{9}\right)\sqrt{3}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{370}{3}\right)\sqrt{3}-\dfrac{43}{8}\right)\right)+\left(\left(\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\dfrac{23}{3}-\left(\left(-15\right)\sqrt{3}\right)\right)-\left(38-\left(\left(\dfrac{115}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(-12+\left(-9\right)\sqrt{3}\right)\right)\\
&=&\left(\left(\dfrac{155}{6}\right)\sqrt{3}\right)-\left(\left(\dfrac{275}{9}\right)\sqrt{3}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{370}{3}\right)\sqrt{3}-\dfrac{43}{8}\right)+\left(\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\dfrac{23}{3}-\left(\left(-15\right)\sqrt{3}\right)\right)-\left(38-\left(\left(\dfrac{115}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(-12+\left(-9\right)\sqrt{3}\right)\\
&=&\left(-\dfrac{5633}{63}\right)\sqrt{3}-\dfrac{1831}{120}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{31}{6}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{75}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{74}{3}\right)\sqrt{75}-\dfrac{43}{8}\right)\right)-\left(\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{27}\right)-\dfrac{23}{3}-\left(\left(-3\right)\sqrt{75}\right)\right)-\left(\left(\left(\dfrac{38}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(\dfrac{155}{6}\right)\sqrt{3}\right)-\left(\left(\dfrac{275}{9}\right)\sqrt{3}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{370}{3}\right)\sqrt{3}-\dfrac{43}{8}\right)\right)-\left(\left(\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\dfrac{23}{3}-\left(\left(-15\right)\sqrt{3}\right)\right)-\left(38-\left(\left(\dfrac{115}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(-12+\left(-9\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{2305}{18}\right)\sqrt{3}+\dfrac{2209}{120}\right)-\left(\left(\dfrac{541}{14}\right)\sqrt{3}-\dfrac{101}{3}\right)\\
&=&\left(-\dfrac{2305}{18}\right)\sqrt{3}+\dfrac{2209}{120}+\left(-\dfrac{541}{14}\right)\sqrt{3}+\dfrac{101}{3}\\
&=&\left(-\dfrac{10502}{63}\right)\sqrt{3}+\dfrac{2083}{40}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{31}{6}\right)\sqrt{75}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{75}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{74}{3}\right)\sqrt{75}-\dfrac{43}{8}\right)\right)\times\left(\left(\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-\dfrac{10}{3}\right)\sqrt{27}\right)-\dfrac{23}{3}-\left(\left(-3\right)\sqrt{75}\right)\right)-\left(\left(\left(\dfrac{38}{3}\right)\sqrt{9}\right)-\left(\left(\dfrac{23}{2}\right)\sqrt{75}\right)-\left(\left(-\dfrac{74}{7}\right)\sqrt{75}\right)\right)-\left(\left(-\dfrac{1}{7}\right)\sqrt{12}\right)-\left(\left(-4\right)\sqrt{9}+\left(-3\right)\sqrt{27}\right)\right)\\
&=&\left(\left(\left(\dfrac{155}{6}\right)\sqrt{3}\right)-\left(\left(\dfrac{275}{9}\right)\sqrt{3}+\dfrac{13}{6}\right)-\left(-\dfrac{76}{5}+\left(\dfrac{370}{3}\right)\sqrt{3}-\dfrac{43}{8}\right)\right)\times\left(\left(\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(\left(-10\right)\sqrt{3}\right)-\dfrac{23}{3}-\left(\left(-15\right)\sqrt{3}\right)\right)-\left(38-\left(\left(\dfrac{115}{2}\right)\sqrt{3}\right)-\left(\left(-\dfrac{370}{7}\right)\sqrt{3}\right)\right)-\left(\left(-\dfrac{2}{7}\right)\sqrt{3}\right)-\left(-12+\left(-9\right)\sqrt{3}\right)\right)\\
&=&\left(\left(-\dfrac{2305}{18}\right)\sqrt{3}+\dfrac{2209}{120}\right)\left(\left(\dfrac{541}{14}\right)\sqrt{3}-\dfrac{101}{3}\right)\\
&=&\left(-\dfrac{1247005}{252}\right)\sqrt{9}+\left(\dfrac{75941021}{15120}\right)\sqrt{3}-\dfrac{223109}{360}\\
&=&-\dfrac{38971913}{2520}+\left(\dfrac{75941021}{15120}\right)\sqrt{3}\\
\end{eqnarray*}