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 \( X=\left(\left(\dfrac{23}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{49}\right)+\left(\left(-\dfrac{26}{3}\right)\sqrt{49}\right)+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}+\left(-\dfrac{59}{4}\right)\sqrt{49}+\left(\left(-\dfrac{14}{9}\right)\sqrt{63}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)+\left(\dfrac{12}{5}\right)\sqrt{49}+\left(-\dfrac{19}{8}\right)\sqrt{63}+\left(\dfrac{46}{5}\right)\sqrt{49}\) et \( Y=\left(-\dfrac{41}{4}\right)\sqrt{49}\) . 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{23}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{49}\right)+\left(\left(-\dfrac{26}{3}\right)\sqrt{49}\right)+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}+\left(-\dfrac{59}{4}\right)\sqrt{49}+\left(\left(-\dfrac{14}{9}\right)\sqrt{63}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)+\left(\dfrac{12}{5}\right)\sqrt{49}+\left(-\dfrac{19}{8}\right)\sqrt{63}+\left(\dfrac{46}{5}\right)\sqrt{49}\right)+\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)\\
&=&\left(\left(\left(\dfrac{115}{2}\right)\sqrt{7}\right)+\dfrac{490}{9}-\dfrac{182}{3}+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}-\dfrac{413}{4}+\left(\left(-\dfrac{14}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)+\dfrac{84}{5}+\left(-\dfrac{57}{8}\right)\sqrt{7}+\dfrac{322}{5}\right)+\left(-\dfrac{287}{4}\right)\\
&=&\left(\left(\dfrac{115}{2}\right)\sqrt{7}\right)+\dfrac{490}{9}-\dfrac{182}{3}+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}-\dfrac{413}{4}+\left(\left(-\dfrac{14}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)+\dfrac{84}{5}+\left(-\dfrac{57}{8}\right)\sqrt{7}+\dfrac{322}{5}-\dfrac{287}{4}\\
&=&\left(\dfrac{677}{24}\right)\sqrt{7}-\dfrac{4141}{45}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{23}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{49}\right)+\left(\left(-\dfrac{26}{3}\right)\sqrt{49}\right)+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}+\left(-\dfrac{59}{4}\right)\sqrt{49}+\left(\left(-\dfrac{14}{9}\right)\sqrt{63}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)+\left(\dfrac{12}{5}\right)\sqrt{49}+\left(-\dfrac{19}{8}\right)\sqrt{63}+\left(\dfrac{46}{5}\right)\sqrt{49}\right)-\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)\\
&=&\left(\left(\left(\dfrac{115}{2}\right)\sqrt{7}\right)+\dfrac{490}{9}-\dfrac{182}{3}+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}-\dfrac{413}{4}+\left(\left(-\dfrac{14}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)+\dfrac{84}{5}+\left(-\dfrac{57}{8}\right)\sqrt{7}+\dfrac{322}{5}\right)-\left(-\dfrac{287}{4}\right)\\
&=&\left(\left(\dfrac{677}{24}\right)\sqrt{7}-\dfrac{3649}{180}\right)-\left(-\dfrac{287}{4}\right)\\
&=&\left(\dfrac{677}{24}\right)\sqrt{7}-\dfrac{3649}{180}+\dfrac{287}{4}\\
&=&\left(\dfrac{677}{24}\right)\sqrt{7}+\dfrac{4633}{90}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{23}{2}\right)\sqrt{175}\right)-\left(\left(-\dfrac{70}{9}\right)\sqrt{49}\right)+\left(\left(-\dfrac{26}{3}\right)\sqrt{49}\right)+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}+\left(-\dfrac{59}{4}\right)\sqrt{49}+\left(\left(-\dfrac{14}{9}\right)\sqrt{63}\right)-\left(\left(\dfrac{7}{2}\right)\sqrt{175}\right)+\left(\dfrac{12}{5}\right)\sqrt{49}+\left(-\dfrac{19}{8}\right)\sqrt{63}+\left(\dfrac{46}{5}\right)\sqrt{49}\right)\times\left(\left(-\dfrac{41}{4}\right)\sqrt{49}\right)\\
&=&\left(\left(\left(\dfrac{115}{2}\right)\sqrt{7}\right)+\dfrac{490}{9}-\dfrac{182}{3}+\dfrac{27}{4}+\dfrac{31}{8}-\dfrac{21}{8}-\dfrac{413}{4}+\left(\left(-\dfrac{14}{3}\right)\sqrt{7}\right)-\left(\left(\dfrac{35}{2}\right)\sqrt{7}\right)+\dfrac{84}{5}+\left(-\dfrac{57}{8}\right)\sqrt{7}+\dfrac{322}{5}\right)\times\left(-\dfrac{287}{4}\right)\\
&=&\left(\left(\dfrac{677}{24}\right)\sqrt{7}-\dfrac{3649}{180}\right)\left(-\dfrac{287}{4}\right)\\
&=&\left(-\dfrac{194299}{96}\right)\sqrt{7}+\dfrac{1047263}{720}\\
&=&\left(-\dfrac{194299}{96}\right)\sqrt{7}+\dfrac{1047263}{720}\\
\end{eqnarray*}