L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=0-\left(\left(-\dfrac{31}{4}\right)\sqrt{63}+\left(\dfrac{15}{4}\right)\sqrt{28}+\left(\dfrac{55}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{28}\right)\) et \( Y=\left(\left(\dfrac{77}{5}\right)\sqrt{175}+\left(-\dfrac{19}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{63}+\left(-\dfrac{5}{3}\right)\sqrt{28}+\left(-7\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{11}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{59}{6}\right)\sqrt{28}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(0-\left(\left(-\dfrac{31}{4}\right)\sqrt{63}+\left(\dfrac{15}{4}\right)\sqrt{28}+\left(\dfrac{55}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{28}\right)\right)+\left(\left(\left(\dfrac{77}{5}\right)\sqrt{175}+\left(-\dfrac{19}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{63}+\left(-\dfrac{5}{3}\right)\sqrt{28}+\left(-7\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{11}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{59}{6}\right)\sqrt{28}\right)\right)\right)\\
&=&\left(0-\left(\left(-\dfrac{93}{4}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{165}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{7}\right)\right)+\left(\left(\left(77\right)\sqrt{7}-\dfrac{133}{2}\right)-\left(\left(\dfrac{171}{4}\right)\sqrt{7}+\left(-\dfrac{10}{3}\right)\sqrt{7}+\left(-21\right)\sqrt{7}\right)-\left(-\dfrac{77}{9}-\left(\left(-\dfrac{59}{3}\right)\sqrt{7}\right)\right)\right)\\
&=&0-\left(\left(-\dfrac{93}{4}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{165}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{7}\right)+\left(\left(77\right)\sqrt{7}-\dfrac{133}{2}\right)-\left(\left(\dfrac{171}{4}\right)\sqrt{7}+\left(-\dfrac{10}{3}\right)\sqrt{7}+\left(-21\right)\sqrt{7}\right)-\left(-\dfrac{77}{9}-\left(\left(-\dfrac{59}{3}\right)\sqrt{7}\right)\right)\\
&=&-\dfrac{1043}{18}+\left(\dfrac{1727}{84}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(0-\left(\left(-\dfrac{31}{4}\right)\sqrt{63}+\left(\dfrac{15}{4}\right)\sqrt{28}+\left(\dfrac{55}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{28}\right)\right)-\left(\left(\left(\dfrac{77}{5}\right)\sqrt{175}+\left(-\dfrac{19}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{63}+\left(-\dfrac{5}{3}\right)\sqrt{28}+\left(-7\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{11}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{59}{6}\right)\sqrt{28}\right)\right)\right)\\
&=&\left(0-\left(\left(-\dfrac{93}{4}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{165}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{7}\right)\right)-\left(\left(\left(77\right)\sqrt{7}-\dfrac{133}{2}\right)-\left(\left(\dfrac{171}{4}\right)\sqrt{7}+\left(-\dfrac{10}{3}\right)\sqrt{7}+\left(-21\right)\sqrt{7}\right)-\left(-\dfrac{77}{9}-\left(\left(-\dfrac{59}{3}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(0+\left(-\dfrac{257}{14}\right)\sqrt{7}\right)-\left(\left(\dfrac{467}{12}\right)\sqrt{7}-\dfrac{1043}{18}\right)\\
&=&0+\left(-\dfrac{257}{14}\right)\sqrt{7}+\left(-\dfrac{467}{12}\right)\sqrt{7}+\dfrac{1043}{18}\\
&=&\dfrac{1043}{18}+\left(-\dfrac{4811}{84}\right)\sqrt{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(0-\left(\left(-\dfrac{31}{4}\right)\sqrt{63}+\left(\dfrac{15}{4}\right)\sqrt{28}+\left(\dfrac{55}{4}\right)\sqrt{63}\right)-\left(\left(-\dfrac{25}{7}\right)\sqrt{28}\right)\right)\times\left(\left(\left(\dfrac{77}{5}\right)\sqrt{175}+\left(-\dfrac{19}{2}\right)\sqrt{49}\right)-\left(\left(\dfrac{57}{4}\right)\sqrt{63}+\left(-\dfrac{5}{3}\right)\sqrt{28}+\left(-7\right)\sqrt{63}\right)-\left(\left(\left(-\dfrac{11}{9}\right)\sqrt{49}\right)-\left(\left(-\dfrac{59}{6}\right)\sqrt{28}\right)\right)\right)\\
&=&\left(0-\left(\left(-\dfrac{93}{4}\right)\sqrt{7}+\left(\dfrac{15}{2}\right)\sqrt{7}+\left(\dfrac{165}{4}\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{7}\right)\sqrt{7}\right)\right)\times\left(\left(\left(77\right)\sqrt{7}-\dfrac{133}{2}\right)-\left(\left(\dfrac{171}{4}\right)\sqrt{7}+\left(-\dfrac{10}{3}\right)\sqrt{7}+\left(-21\right)\sqrt{7}\right)-\left(-\dfrac{77}{9}-\left(\left(-\dfrac{59}{3}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(0+\left(-\dfrac{257}{14}\right)\sqrt{7}\right)\left(\left(\dfrac{467}{12}\right)\sqrt{7}-\dfrac{1043}{18}\right)\\
&=&\left(\dfrac{38293}{36}\right)\sqrt{7}+0+\left(-\dfrac{120019}{168}\right)\sqrt{49}\\
&=&\left(\dfrac{38293}{36}\right)\sqrt{7}-\dfrac{120019}{24}\\
\end{eqnarray*}