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 \( X=\dfrac{67}{7}-\left(\left(\dfrac{69}{7}\right)\sqrt{63}+\left(-3\right)\sqrt{49}+\dfrac{40}{7}\right)\) et \( Y=\left(\left(-\dfrac{3}{2}\right)\sqrt{63}+\left(-\dfrac{51}{5}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{28}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{11}{7}\right)\sqrt{175}\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{67}{7}-\left(\left(\dfrac{69}{7}\right)\sqrt{63}+\left(-3\right)\sqrt{49}+\dfrac{40}{7}\right)\right)+\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{63}+\left(-\dfrac{51}{5}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{28}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{11}{7}\right)\sqrt{175}\right)\right)\\
&=&\left(\dfrac{67}{7}-\left(\left(\dfrac{207}{7}\right)\sqrt{7}-21+\dfrac{40}{7}\right)\right)+\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\dfrac{16}{9}\right)\sqrt{7}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{7}\right)\right)\\
&=&\dfrac{67}{7}-\left(\left(\dfrac{207}{7}\right)\sqrt{7}-21+\dfrac{40}{7}\right)+\left(\left(-\dfrac{9}{2}\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\dfrac{16}{9}\right)\sqrt{7}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{7}\right)\\
&=&\dfrac{235}{21}+\left(-\dfrac{9505}{126}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{67}{7}-\left(\left(\dfrac{69}{7}\right)\sqrt{63}+\left(-3\right)\sqrt{49}+\dfrac{40}{7}\right)\right)-\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{63}+\left(-\dfrac{51}{5}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{28}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{11}{7}\right)\sqrt{175}\right)\right)\\
&=&\left(\dfrac{67}{7}-\left(\left(\dfrac{207}{7}\right)\sqrt{7}-21+\dfrac{40}{7}\right)\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\dfrac{16}{9}\right)\sqrt{7}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{7}\right)\right)\\
&=&\left(\dfrac{174}{7}+\left(-\dfrac{207}{7}\right)\sqrt{7}\right)-\left(\left(-\dfrac{5779}{126}\right)\sqrt{7}-\dfrac{41}{3}\right)\\
&=&\dfrac{174}{7}+\left(-\dfrac{207}{7}\right)\sqrt{7}+\left(\dfrac{5779}{126}\right)\sqrt{7}+\dfrac{41}{3}\\
&=&\dfrac{809}{21}+\left(\dfrac{2053}{126}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{67}{7}-\left(\left(\dfrac{69}{7}\right)\sqrt{63}+\left(-3\right)\sqrt{49}+\dfrac{40}{7}\right)\right)\times\left(\left(\left(-\dfrac{3}{2}\right)\sqrt{63}+\left(-\dfrac{51}{5}\right)\sqrt{175}+\left(\dfrac{8}{9}\right)\sqrt{28}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{11}{7}\right)\sqrt{175}\right)\right)\\
&=&\left(\dfrac{67}{7}-\left(\left(\dfrac{207}{7}\right)\sqrt{7}-21+\dfrac{40}{7}\right)\right)\times\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{7}+\left(-51\right)\sqrt{7}+\left(\dfrac{16}{9}\right)\sqrt{7}-9-\dfrac{14}{3}\right)-\left(\left(-\dfrac{55}{7}\right)\sqrt{7}\right)\right)\\
&=&\left(\dfrac{174}{7}+\left(-\dfrac{207}{7}\right)\sqrt{7}\right)\left(\left(-\dfrac{5779}{126}\right)\sqrt{7}-\dfrac{41}{3}\right)\\
&=&\left(-\dfrac{108182}{147}\right)\sqrt{7}-\dfrac{2378}{7}+\left(\dfrac{132917}{98}\right)\sqrt{49}\\
&=&\left(-\dfrac{108182}{147}\right)\sqrt{7}+\dfrac{128161}{14}\\
\end{eqnarray*}