L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.
Exercice
Soit \( X=\left(5+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{14}{5}\right)\sqrt{49}\right)-\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)-\left(\left(8\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{25}{6}\right)\sqrt{175}\right)\) et \( Y=\left(\left(\dfrac{17}{2}\right)\sqrt{175}+\left(\dfrac{53}{7}\right)\sqrt{49}+\left(-\dfrac{9}{2}\right)\sqrt{49}\right)+\dfrac{26}{7}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(5+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{14}{5}\right)\sqrt{49}\right)-\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)-\left(\left(8\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{25}{6}\right)\sqrt{175}\right)\right)+\left(\left(\left(\dfrac{17}{2}\right)\sqrt{175}+\left(\dfrac{53}{7}\right)\sqrt{49}+\left(-\dfrac{9}{2}\right)\sqrt{49}\right)+\dfrac{26}{7}\right)\\
&=&\left(\left(5+\dfrac{161}{3}-\dfrac{98}{5}\right)-\left(-\dfrac{392}{9}-\left(\left(-\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)-\left(\left(24\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{125}{6}\right)\sqrt{7}\right)\right)+\left(\left(\left(\dfrac{85}{2}\right)\sqrt{7}+53-\dfrac{63}{2}\right)+\dfrac{26}{7}\right)\\
&=&\left(5+\dfrac{161}{3}-\dfrac{98}{5}\right)-\left(-\dfrac{392}{9}-\left(\left(-\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)-\left(\left(24\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{125}{6}\right)\sqrt{7}\right)+\left(\left(\dfrac{85}{2}\right)\sqrt{7}+53-\dfrac{63}{2}\right)+\dfrac{26}{7}\\
&=&\dfrac{67937}{630}+\left(\dfrac{326}{3}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(5+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{14}{5}\right)\sqrt{49}\right)-\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)-\left(\left(8\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{25}{6}\right)\sqrt{175}\right)\right)-\left(\left(\left(\dfrac{17}{2}\right)\sqrt{175}+\left(\dfrac{53}{7}\right)\sqrt{49}+\left(-\dfrac{9}{2}\right)\sqrt{49}\right)+\dfrac{26}{7}\right)\\
&=&\left(\left(5+\dfrac{161}{3}-\dfrac{98}{5}\right)-\left(-\dfrac{392}{9}-\left(\left(-\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)-\left(\left(24\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{125}{6}\right)\sqrt{7}\right)\right)-\left(\left(\left(\dfrac{85}{2}\right)\sqrt{7}+53-\dfrac{63}{2}\right)+\dfrac{26}{7}\right)\\
&=&\left(\dfrac{3718}{45}+\left(\dfrac{397}{6}\right)\sqrt{7}\right)-\left(\left(\dfrac{85}{2}\right)\sqrt{7}+\dfrac{353}{14}\right)\\
&=&\dfrac{3718}{45}+\left(\dfrac{397}{6}\right)\sqrt{7}+\left(-\dfrac{85}{2}\right)\sqrt{7}-\dfrac{353}{14}\\
&=&\dfrac{36167}{630}+\left(\dfrac{71}{3}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(5+\left(\dfrac{23}{3}\right)\sqrt{49}+\left(-\dfrac{14}{5}\right)\sqrt{49}\right)-\left(\left(\left(-\dfrac{56}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{28}\right)-\left(\left(\dfrac{31}{2}\right)\sqrt{28}\right)-\left(\left(8\right)\sqrt{63}\right)\right)-\left(\left(-\dfrac{25}{6}\right)\sqrt{175}\right)\right)\times\left(\left(\left(\dfrac{17}{2}\right)\sqrt{175}+\left(\dfrac{53}{7}\right)\sqrt{49}+\left(-\dfrac{9}{2}\right)\sqrt{49}\right)+\dfrac{26}{7}\right)\\
&=&\left(\left(5+\dfrac{161}{3}-\dfrac{98}{5}\right)-\left(-\dfrac{392}{9}-\left(\left(-\dfrac{29}{3}\right)\sqrt{7}\right)-\left(\left(31\right)\sqrt{7}\right)-\left(\left(24\right)\sqrt{7}\right)\right)-\left(\left(-\dfrac{125}{6}\right)\sqrt{7}\right)\right)\times\left(\left(\left(\dfrac{85}{2}\right)\sqrt{7}+53-\dfrac{63}{2}\right)+\dfrac{26}{7}\right)\\
&=&\left(\dfrac{3718}{45}+\left(\dfrac{397}{6}\right)\sqrt{7}\right)\left(\left(\dfrac{85}{2}\right)\sqrt{7}+\dfrac{353}{14}\right)\\
&=&\left(\dfrac{1305307}{252}\right)\sqrt{7}+\dfrac{656227}{315}+\left(\dfrac{33745}{12}\right)\sqrt{49}\\
&=&\left(\dfrac{1305307}{252}\right)\sqrt{7}+\dfrac{27427483}{1260}\\
\end{eqnarray*}