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 \( X=\left(\left(-\dfrac{22}{3}\right)\sqrt{49}+\left(\dfrac{26}{3}\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{5}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{63}\right)\right)-\dfrac{65}{6}\) et \( Y=\left(\left(-\dfrac{64}{3}\right)\sqrt{175}+\left(-\dfrac{64}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{28}+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{23}{3}\right)\sqrt{175}\right)-\dfrac{41}{6}\) . 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{22}{3}\right)\sqrt{49}+\left(\dfrac{26}{3}\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{5}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{63}\right)\right)-\dfrac{65}{6}\right)+\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{175}+\left(-\dfrac{64}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{28}+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{23}{3}\right)\sqrt{175}\right)-\dfrac{41}{6}\right)\\
&=&\left(\left(-\dfrac{154}{3}+\left(\dfrac{52}{3}\right)\sqrt{7}\right)-\left(\dfrac{35}{4}-\left(\left(\dfrac{177}{4}\right)\sqrt{7}\right)\right)-\dfrac{65}{6}\right)+\left(\left(\left(-\dfrac{320}{3}\right)\sqrt{7}+\left(-\dfrac{320}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{61}{4}\right)\sqrt{7}+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{115}{3}\right)\sqrt{7}\right)-\dfrac{41}{6}\right)\\
&=&\left(-\dfrac{154}{3}+\left(\dfrac{52}{3}\right)\sqrt{7}\right)-\left(\dfrac{35}{4}-\left(\left(\dfrac{177}{4}\right)\sqrt{7}\right)\right)-\dfrac{65}{6}+\left(\left(-\dfrac{320}{3}\right)\sqrt{7}+\left(-\dfrac{320}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{61}{4}\right)\sqrt{7}+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{115}{3}\right)\sqrt{7}\right)-\dfrac{41}{6}\\
&=&-\dfrac{311}{4}+\left(-\dfrac{1253}{12}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{49}+\left(\dfrac{26}{3}\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{5}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{63}\right)\right)-\dfrac{65}{6}\right)-\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{175}+\left(-\dfrac{64}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{28}+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{23}{3}\right)\sqrt{175}\right)-\dfrac{41}{6}\right)\\
&=&\left(\left(-\dfrac{154}{3}+\left(\dfrac{52}{3}\right)\sqrt{7}\right)-\left(\dfrac{35}{4}-\left(\left(\dfrac{177}{4}\right)\sqrt{7}\right)\right)-\dfrac{65}{6}\right)-\left(\left(\left(-\dfrac{320}{3}\right)\sqrt{7}+\left(-\dfrac{320}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{61}{4}\right)\sqrt{7}+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{115}{3}\right)\sqrt{7}\right)-\dfrac{41}{6}\right)\\
&=&\left(-\dfrac{851}{12}+\left(\dfrac{739}{12}\right)\sqrt{7}\right)-\left(\left(-166\right)\sqrt{7}-\dfrac{41}{6}\right)\\
&=&-\dfrac{851}{12}+\left(\dfrac{739}{12}\right)\sqrt{7}+\left(166\right)\sqrt{7}+\dfrac{41}{6}\\
&=&-\dfrac{769}{12}+\left(\dfrac{2731}{12}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{22}{3}\right)\sqrt{49}+\left(\dfrac{26}{3}\right)\sqrt{28}\right)-\left(\left(\left(\dfrac{5}{4}\right)\sqrt{49}\right)-\left(\left(\dfrac{59}{4}\right)\sqrt{63}\right)\right)-\dfrac{65}{6}\right)\times\left(\left(\left(-\dfrac{64}{3}\right)\sqrt{175}+\left(-\dfrac{64}{3}\right)\sqrt{175}\right)-\left(\left(-\dfrac{61}{8}\right)\sqrt{28}+\left(\dfrac{5}{4}\right)\sqrt{175}+\left(-\dfrac{23}{3}\right)\sqrt{175}\right)-\dfrac{41}{6}\right)\\
&=&\left(\left(-\dfrac{154}{3}+\left(\dfrac{52}{3}\right)\sqrt{7}\right)-\left(\dfrac{35}{4}-\left(\left(\dfrac{177}{4}\right)\sqrt{7}\right)\right)-\dfrac{65}{6}\right)\times\left(\left(\left(-\dfrac{320}{3}\right)\sqrt{7}+\left(-\dfrac{320}{3}\right)\sqrt{7}\right)-\left(\left(-\dfrac{61}{4}\right)\sqrt{7}+\left(\dfrac{25}{4}\right)\sqrt{7}+\left(-\dfrac{115}{3}\right)\sqrt{7}\right)-\dfrac{41}{6}\right)\\
&=&\left(-\dfrac{851}{12}+\left(\dfrac{739}{12}\right)\sqrt{7}\right)\left(\left(-166\right)\sqrt{7}-\dfrac{41}{6}\right)\\
&=&\left(\dfrac{817297}{72}\right)\sqrt{7}+\dfrac{34891}{72}+\left(-\dfrac{61337}{6}\right)\sqrt{49}\\
&=&\left(\dfrac{817297}{72}\right)\sqrt{7}-\dfrac{5117417}{72}\\
\end{eqnarray*}