L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\dfrac{17}{3}\right)\sqrt{28}\) et \( Y=\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{81}{2}\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{17}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{49}{8}\right)\sqrt{28}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-1\right)\sqrt{63}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\dfrac{17}{3}\right)\sqrt{28}\right)+\left(\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{81}{2}\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{17}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{49}{8}\right)\sqrt{28}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-1\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\dfrac{34}{3}\right)\sqrt{7}\right)+\left(\dfrac{189}{2}-\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{3}\right)\sqrt{7}\right)+\dfrac{567}{2}\right)-\left(\left(\left(-\dfrac{34}{3}\right)\sqrt{7}\right)-\dfrac{42}{5}-\dfrac{189}{2}-\left(\left(-\dfrac{49}{4}\right)\sqrt{7}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-3\right)\sqrt{7}\right)\right)\\
&=&\left(\dfrac{34}{3}\right)\sqrt{7}+\dfrac{189}{2}-\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{3}\right)\sqrt{7}\right)+\dfrac{567}{2}\right)-\left(\left(\left(-\dfrac{34}{3}\right)\sqrt{7}\right)-\dfrac{42}{5}-\dfrac{189}{2}-\left(\left(-\dfrac{49}{4}\right)\sqrt{7}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-3\right)\sqrt{7}\right)\\
&=&\left(\dfrac{241}{12}\right)\sqrt{7}-\dfrac{21971}{210}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{17}{3}\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{81}{2}\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{17}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{49}{8}\right)\sqrt{28}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-1\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\dfrac{34}{3}\right)\sqrt{7}\right)-\left(\dfrac{189}{2}-\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{3}\right)\sqrt{7}\right)+\dfrac{567}{2}\right)-\left(\left(\left(-\dfrac{34}{3}\right)\sqrt{7}\right)-\dfrac{42}{5}-\dfrac{189}{2}-\left(\left(-\dfrac{49}{4}\right)\sqrt{7}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-3\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{34}{3}\right)\sqrt{7}\right)-\left(-\dfrac{21971}{210}+\left(\dfrac{35}{4}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{34}{3}\right)\sqrt{7}+\dfrac{21971}{210}+\left(-\dfrac{35}{4}\right)\sqrt{7}\\
&=&\left(\dfrac{31}{12}\right)\sqrt{7}+\dfrac{21971}{210}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{17}{3}\right)\sqrt{28}\right)\times\left(\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(\left(-8\right)\sqrt{175}\right)-\left(\left(-\dfrac{20}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{81}{2}\right)\sqrt{49}\right)\right)-\left(\left(\left(-\dfrac{17}{3}\right)\sqrt{28}\right)-\left(\left(\dfrac{6}{5}\right)\sqrt{49}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{49}\right)-\left(\left(-\dfrac{49}{8}\right)\sqrt{28}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-1\right)\sqrt{63}\right)\right)\\
&=&\left(\left(\dfrac{34}{3}\right)\sqrt{7}\right)\times\left(\dfrac{189}{2}-\left(\left(\left(-40\right)\sqrt{7}\right)-\left(\left(-\dfrac{100}{3}\right)\sqrt{7}\right)+\dfrac{567}{2}\right)-\left(\left(\left(-\dfrac{34}{3}\right)\sqrt{7}\right)-\dfrac{42}{5}-\dfrac{189}{2}-\left(\left(-\dfrac{49}{4}\right)\sqrt{7}\right)\right)-\left(\dfrac{6}{7}+\dfrac{53}{3}+\left(-3\right)\sqrt{7}\right)\right)\\
&=&\left(\left(\dfrac{34}{3}\right)\sqrt{7}\right)\left(-\dfrac{21971}{210}+\left(\dfrac{35}{4}\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{373507}{315}\right)\sqrt{7}+\left(\dfrac{595}{6}\right)\sqrt{49}\\
&=&\left(-\dfrac{373507}{315}\right)\sqrt{7}+\dfrac{4165}{6}\\
\end{eqnarray*}