L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{63}\right)\right)-\left(\dfrac{32}{3}+\left(-9\right)\sqrt{63}\right)-\left(\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{28}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{63}\right)\right)\) et \( Y=\left(\dfrac{46}{7}\right)\sqrt{49}+\left(\dfrac{78}{7}\right)\sqrt{175}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-\dfrac{9}{2}\right)\sqrt{28}\) . 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{25}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{63}\right)\right)-\left(\dfrac{32}{3}+\left(-9\right)\sqrt{63}\right)-\left(\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{28}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{63}\right)\right)\right)+\left(\left(\dfrac{46}{7}\right)\sqrt{49}+\left(\dfrac{78}{7}\right)\sqrt{175}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-\dfrac{9}{2}\right)\sqrt{28}\right)\\
&=&\left(\left(-\dfrac{175}{4}-\left(\left(-\dfrac{50}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{343}{5}-\left(\left(-\dfrac{68}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{32}{3}+\left(-27\right)\sqrt{7}\right)-\left(\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{128}{3}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-50\right)\sqrt{7}\right)\right)\right)+\left(46+\left(\dfrac{390}{7}\right)\sqrt{7}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-9\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{175}{4}-\left(\left(-\dfrac{50}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{343}{5}-\left(\left(-\dfrac{68}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{32}{3}+\left(-27\right)\sqrt{7}\right)-\left(\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{128}{3}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-50\right)\sqrt{7}\right)\right)+46+\left(\dfrac{390}{7}\right)\sqrt{7}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-9\right)\sqrt{7}\\
&=&-\dfrac{8758}{105}+\left(-\dfrac{524}{21}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{63}\right)\right)-\left(\dfrac{32}{3}+\left(-9\right)\sqrt{63}\right)-\left(\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{28}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{63}\right)\right)\right)-\left(\left(\dfrac{46}{7}\right)\sqrt{49}+\left(\dfrac{78}{7}\right)\sqrt{175}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-\dfrac{9}{2}\right)\sqrt{28}\right)\\
&=&\left(\left(-\dfrac{175}{4}-\left(\left(-\dfrac{50}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{343}{5}-\left(\left(-\dfrac{68}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{32}{3}+\left(-27\right)\sqrt{7}\right)-\left(\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{128}{3}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-50\right)\sqrt{7}\right)\right)\right)-\left(46+\left(\dfrac{390}{7}\right)\sqrt{7}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-9\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{7381}{60}+\left(-\dfrac{215}{3}\right)\sqrt{7}\right)-\left(\dfrac{1109}{28}+\left(\dfrac{327}{7}\right)\sqrt{7}\right)\\
&=&-\dfrac{7381}{60}+\left(-\dfrac{215}{3}\right)\sqrt{7}+-\dfrac{1109}{28}+\left(-\dfrac{327}{7}\right)\sqrt{7}\\
&=&-\dfrac{34151}{210}+\left(-\dfrac{2486}{21}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(-\dfrac{25}{4}\right)\sqrt{49}\right)-\left(\left(-\dfrac{25}{3}\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{49}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{68}{9}\right)\sqrt{63}\right)\right)-\left(\dfrac{32}{3}+\left(-9\right)\sqrt{63}\right)-\left(\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{64}{3}\right)\sqrt{28}\right)-\left(\left(5\right)\sqrt{28}\right)-\left(\left(-\dfrac{50}{3}\right)\sqrt{63}\right)\right)\right)\times\left(\left(\dfrac{46}{7}\right)\sqrt{49}+\left(\dfrac{78}{7}\right)\sqrt{175}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-\dfrac{9}{2}\right)\sqrt{28}\right)\\
&=&\left(\left(-\dfrac{175}{4}-\left(\left(-\dfrac{50}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{343}{5}-\left(\left(-\dfrac{68}{3}\right)\sqrt{7}\right)\right)-\left(\dfrac{32}{3}+\left(-27\right)\sqrt{7}\right)-\left(\left(\left(10\right)\sqrt{7}\right)-\left(\left(-\dfrac{128}{3}\right)\sqrt{7}\right)-\left(\left(10\right)\sqrt{7}\right)-\left(\left(-50\right)\sqrt{7}\right)\right)\right)\times\left(46+\left(\dfrac{390}{7}\right)\sqrt{7}-\dfrac{71}{7}+\dfrac{15}{4}+\left(-9\right)\sqrt{7}\right)\\
&=&\left(-\dfrac{7381}{60}+\left(-\dfrac{215}{3}\right)\sqrt{7}\right)\left(\dfrac{1109}{28}+\left(\dfrac{327}{7}\right)\sqrt{7}\right)\\
&=&-\dfrac{8185529}{1680}+\left(-\dfrac{1802881}{210}\right)\sqrt{7}+\left(-\dfrac{23435}{7}\right)\sqrt{49}\\
&=&-\dfrac{47556329}{1680}+\left(-\dfrac{1802881}{210}\right)\sqrt{7}\\
\end{eqnarray*}