L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{39}{8}\right)\sqrt{75}+\left(5\right)\sqrt{75}-\dfrac{13}{7}+\left(1\right)\sqrt{9}+\left(-\dfrac{47}{6}\right)\sqrt{12}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{27}\right)+1\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)\right)\) et \( Y=\left(-\dfrac{7}{3}\right)\sqrt{12}+\left(-\dfrac{76}{3}\right)\sqrt{12}+\left(0\right)\sqrt{75}+\left(\dfrac{75}{8}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{9}+\left(-\dfrac{13}{2}\right)\sqrt{12}+\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(\dfrac{13}{9}\right)\sqrt{9}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{75}+\left(5\right)\sqrt{75}-\dfrac{13}{7}+\left(1\right)\sqrt{9}+\left(-\dfrac{47}{6}\right)\sqrt{12}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{27}\right)+1\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)\right)\right)+\left(\left(-\dfrac{7}{3}\right)\sqrt{12}+\left(-\dfrac{76}{3}\right)\sqrt{12}+\left(0\right)\sqrt{75}+\left(\dfrac{75}{8}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{9}+\left(-\dfrac{13}{2}\right)\sqrt{12}+\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(\dfrac{13}{9}\right)\sqrt{9}\right)\\
&=&\left(\left(\left(-\dfrac{195}{8}\right)\sqrt{3}+\left(25\right)\sqrt{3}-\dfrac{13}{7}+3+\left(-\dfrac{47}{3}\right)\sqrt{3}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{225}{8}\right)\sqrt{3}\right)+1\right)-\left(\left(\left(\dfrac{68}{3}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)\right)+\left(\left(-\dfrac{14}{3}\right)\sqrt{3}+\left(-\dfrac{152}{3}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{225}{8}\right)\sqrt{3}-\dfrac{114}{5}+\left(-13\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{13}{3}\right)\\
&=&\left(\left(-\dfrac{195}{8}\right)\sqrt{3}+\left(25\right)\sqrt{3}-\dfrac{13}{7}+3+\left(-\dfrac{47}{3}\right)\sqrt{3}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{225}{8}\right)\sqrt{3}\right)+1\right)-\left(\left(\left(\dfrac{68}{3}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)+\left(-\dfrac{14}{3}\right)\sqrt{3}+\left(-\dfrac{152}{3}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{225}{8}\right)\sqrt{3}-\dfrac{114}{5}+\left(-13\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{13}{3}\\
&=&\left(-\dfrac{3049}{24}\right)\sqrt{3}-\dfrac{314}{105}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{75}+\left(5\right)\sqrt{75}-\dfrac{13}{7}+\left(1\right)\sqrt{9}+\left(-\dfrac{47}{6}\right)\sqrt{12}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{27}\right)+1\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)\right)\right)-\left(\left(-\dfrac{7}{3}\right)\sqrt{12}+\left(-\dfrac{76}{3}\right)\sqrt{12}+\left(0\right)\sqrt{75}+\left(\dfrac{75}{8}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{9}+\left(-\dfrac{13}{2}\right)\sqrt{12}+\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(\dfrac{13}{9}\right)\sqrt{9}\right)\\
&=&\left(\left(\left(-\dfrac{195}{8}\right)\sqrt{3}+\left(25\right)\sqrt{3}-\dfrac{13}{7}+3+\left(-\dfrac{47}{3}\right)\sqrt{3}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{225}{8}\right)\sqrt{3}\right)+1\right)-\left(\left(\left(\dfrac{68}{3}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)\right)-\left(\left(-\dfrac{14}{3}\right)\sqrt{3}+\left(-\dfrac{152}{3}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{225}{8}\right)\sqrt{3}-\dfrac{114}{5}+\left(-13\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{13}{3}\right)\\
&=&\left(\left(-\dfrac{383}{6}\right)\sqrt{3}+\dfrac{325}{21}\right)-\left(\left(-\dfrac{1517}{24}\right)\sqrt{3}-\dfrac{277}{15}\right)\\
&=&\left(-\dfrac{383}{6}\right)\sqrt{3}+\dfrac{325}{21}+\left(\dfrac{1517}{24}\right)\sqrt{3}+\dfrac{277}{15}\\
&=&\left(-\dfrac{5}{8}\right)\sqrt{3}+\dfrac{1188}{35}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{39}{8}\right)\sqrt{75}+\left(5\right)\sqrt{75}-\dfrac{13}{7}+\left(1\right)\sqrt{9}+\left(-\dfrac{47}{6}\right)\sqrt{12}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{75}{8}\right)\sqrt{27}\right)+1\right)-\left(\left(\left(\dfrac{34}{3}\right)\sqrt{12}\right)-\left(\left(1\right)\sqrt{12}\right)\right)\right)\times\left(\left(-\dfrac{7}{3}\right)\sqrt{12}+\left(-\dfrac{76}{3}\right)\sqrt{12}+\left(0\right)\sqrt{75}+\left(\dfrac{75}{8}\right)\sqrt{27}+\left(-\dfrac{38}{5}\right)\sqrt{9}+\left(-\dfrac{13}{2}\right)\sqrt{12}+\left(-\dfrac{23}{3}\right)\sqrt{27}+\left(\dfrac{13}{9}\right)\sqrt{9}\right)\\
&=&\left(\left(\left(-\dfrac{195}{8}\right)\sqrt{3}+\left(25\right)\sqrt{3}-\dfrac{13}{7}+3+\left(-\dfrac{47}{3}\right)\sqrt{3}\right)-\left(-\dfrac{46}{3}-\left(\left(-\dfrac{225}{8}\right)\sqrt{3}\right)+1\right)-\left(\left(\left(\dfrac{68}{3}\right)\sqrt{3}\right)-\left(\left(2\right)\sqrt{3}\right)\right)\right)\times\left(\left(-\dfrac{14}{3}\right)\sqrt{3}+\left(-\dfrac{152}{3}\right)\sqrt{3}+\left(0\right)\sqrt{3}+\left(\dfrac{225}{8}\right)\sqrt{3}-\dfrac{114}{5}+\left(-13\right)\sqrt{3}+\left(-23\right)\sqrt{3}+\dfrac{13}{3}\right)\\
&=&\left(\left(-\dfrac{383}{6}\right)\sqrt{3}+\dfrac{325}{21}\right)\left(\left(-\dfrac{1517}{24}\right)\sqrt{3}-\dfrac{277}{15}\right)\\
&=&\left(\dfrac{581011}{144}\right)\sqrt{9}+\left(\dfrac{505423}{2520}\right)\sqrt{3}-\dfrac{18005}{63}\\
&=&\dfrac{11913151}{1008}+\left(\dfrac{505423}{2520}\right)\sqrt{3}\\
\end{eqnarray*}