L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\dfrac{5}{6}\right)\sqrt{125}+\left(\dfrac{31}{5}\right)\sqrt{25}+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-9\right)\sqrt{45}\right)\) et \( Y=-\dfrac{26}{5}-\left(\left(-\dfrac{55}{3}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{25}\right)-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-3\right)\sqrt{125}\right)\right)\) . 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{5}{6}\right)\sqrt{125}+\left(\dfrac{31}{5}\right)\sqrt{25}+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-9\right)\sqrt{45}\right)\right)+\left(-\dfrac{26}{5}-\left(\left(-\dfrac{55}{3}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{25}\right)-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-3\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\left(\dfrac{25}{6}\right)\sqrt{5}+31+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-27\right)\sqrt{5}\right)\right)+\left(-\dfrac{26}{5}-\left(\left(-55\right)\sqrt{5}\right)-\left(\dfrac{380}{7}-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-15\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{25}{6}\right)\sqrt{5}+31+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-27\right)\sqrt{5}\right)-\dfrac{26}{5}-\left(\left(-55\right)\sqrt{5}\right)-\left(\dfrac{380}{7}-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-15\right)\sqrt{5}\right)\right)\\
&=&\left(\dfrac{427}{6}\right)\sqrt{5}+\dfrac{13523}{1260}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{5}{6}\right)\sqrt{125}+\left(\dfrac{31}{5}\right)\sqrt{25}+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-9\right)\sqrt{45}\right)\right)-\left(-\dfrac{26}{5}-\left(\left(-\dfrac{55}{3}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{25}\right)-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-3\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\left(\dfrac{25}{6}\right)\sqrt{5}+31+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-27\right)\sqrt{5}\right)\right)-\left(-\dfrac{26}{5}-\left(\left(-55\right)\sqrt{5}\right)-\left(\dfrac{380}{7}-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-15\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{187}{6}\right)\sqrt{5}+\dfrac{5329}{84}\right)-\left(-\dfrac{16603}{315}+\left(40\right)\sqrt{5}\right)\\
&=&\left(\dfrac{187}{6}\right)\sqrt{5}+\dfrac{5329}{84}+\dfrac{16603}{315}+\left(-40\right)\sqrt{5}\\
&=&\left(-\dfrac{53}{6}\right)\sqrt{5}+\dfrac{146347}{1260}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{5}{6}\right)\sqrt{125}+\left(\dfrac{31}{5}\right)\sqrt{25}+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-9\right)\sqrt{45}\right)\right)\times\left(-\dfrac{26}{5}-\left(\left(-\dfrac{55}{3}\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{76}{7}\right)\sqrt{25}\right)-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-3\right)\sqrt{125}\right)\right)\right)\\
&=&\left(\left(\left(\dfrac{25}{6}\right)\sqrt{5}+31+\dfrac{34}{7}-5+\dfrac{81}{4}\right)-\left(-\dfrac{25}{2}-7\right)-\left(\dfrac{43}{6}+\left(-27\right)\sqrt{5}\right)\right)\times\left(-\dfrac{26}{5}-\left(\left(-55\right)\sqrt{5}\right)-\left(\dfrac{380}{7}-\dfrac{26}{3}+\dfrac{17}{9}-\left(\left(-15\right)\sqrt{5}\right)\right)\right)\\
&=&\left(\left(\dfrac{187}{6}\right)\sqrt{5}+\dfrac{5329}{84}\right)\left(-\dfrac{16603}{315}+\left(40\right)\sqrt{5}\right)\\
&=&\left(\dfrac{1691339}{1890}\right)\sqrt{5}+\left(\dfrac{3740}{3}\right)\sqrt{25}-\dfrac{88477387}{26460}\\
&=&\left(\dfrac{1691339}{1890}\right)\sqrt{5}+\dfrac{76456613}{26460}\\
\end{eqnarray*}