L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(-\dfrac{14}{3}\right)\sqrt{28}+\left(\dfrac{59}{7}\right)\sqrt{63}+\left(-\dfrac{19}{4}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{63}+\left(\dfrac{27}{7}\right)\sqrt{175}+\left(-\dfrac{29}{8}\right)\sqrt{175}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(9\right)\sqrt{28}+\left(-\dfrac{7}{4}\right)\sqrt{175}\) et \( Y=\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-\dfrac{15}{2}\right)\sqrt{28}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{175}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{28}+\left(\dfrac{59}{7}\right)\sqrt{63}+\left(-\dfrac{19}{4}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{63}+\left(\dfrac{27}{7}\right)\sqrt{175}+\left(-\dfrac{29}{8}\right)\sqrt{175}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(9\right)\sqrt{28}+\left(-\dfrac{7}{4}\right)\sqrt{175}\right)+\left(\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-\dfrac{15}{2}\right)\sqrt{28}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(-\dfrac{28}{3}\right)\sqrt{7}+\left(\dfrac{177}{7}\right)\sqrt{7}-\dfrac{133}{4}+\left(4\right)\sqrt{7}+\left(\dfrac{135}{7}\right)\sqrt{7}+\left(-\dfrac{145}{8}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(18\right)\sqrt{7}+\left(-\dfrac{35}{4}\right)\sqrt{7}\right)+\left(\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(-\dfrac{28}{3}\right)\sqrt{7}+\left(\dfrac{177}{7}\right)\sqrt{7}-\dfrac{133}{4}+\left(4\right)\sqrt{7}+\left(\dfrac{135}{7}\right)\sqrt{7}+\left(-\dfrac{145}{8}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(18\right)\sqrt{7}+\left(-\dfrac{35}{4}\right)\sqrt{7}+\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{7}\right)\right)\\
&=&\left(\dfrac{11243}{504}\right)\sqrt{7}-\dfrac{75}{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{28}+\left(\dfrac{59}{7}\right)\sqrt{63}+\left(-\dfrac{19}{4}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{63}+\left(\dfrac{27}{7}\right)\sqrt{175}+\left(-\dfrac{29}{8}\right)\sqrt{175}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(9\right)\sqrt{28}+\left(-\dfrac{7}{4}\right)\sqrt{175}\right)-\left(\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-\dfrac{15}{2}\right)\sqrt{28}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(-\dfrac{28}{3}\right)\sqrt{7}+\left(\dfrac{177}{7}\right)\sqrt{7}-\dfrac{133}{4}+\left(4\right)\sqrt{7}+\left(\dfrac{135}{7}\right)\sqrt{7}+\left(-\dfrac{145}{8}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(18\right)\sqrt{7}+\left(-\dfrac{35}{4}\right)\sqrt{7}\right)-\left(\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(\dfrac{7201}{168}\right)\sqrt{7}-\dfrac{133}{4}\right)-\left(-\dfrac{17}{4}+\left(-\dfrac{185}{9}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{7201}{168}\right)\sqrt{7}-\dfrac{133}{4}+\dfrac{17}{4}+\left(\dfrac{185}{9}\right)\sqrt{7}\\
&=&\left(\dfrac{31963}{504}\right)\sqrt{7}-29\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(-\dfrac{14}{3}\right)\sqrt{28}+\left(\dfrac{59}{7}\right)\sqrt{63}+\left(-\dfrac{19}{4}\right)\sqrt{49}+\left(\dfrac{4}{3}\right)\sqrt{63}+\left(\dfrac{27}{7}\right)\sqrt{175}+\left(-\dfrac{29}{8}\right)\sqrt{175}+\left(\dfrac{5}{2}\right)\sqrt{175}+\left(9\right)\sqrt{28}+\left(-\dfrac{7}{4}\right)\sqrt{175}\right)\times\left(\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-\dfrac{15}{2}\right)\sqrt{28}\right)-\left(\left(-\dfrac{10}{9}\right)\sqrt{175}\right)\right)\right)\\
&=&\left(\left(-\dfrac{28}{3}\right)\sqrt{7}+\left(\dfrac{177}{7}\right)\sqrt{7}-\dfrac{133}{4}+\left(4\right)\sqrt{7}+\left(\dfrac{135}{7}\right)\sqrt{7}+\left(-\dfrac{145}{8}\right)\sqrt{7}+\left(\dfrac{25}{2}\right)\sqrt{7}+\left(18\right)\sqrt{7}+\left(-\dfrac{35}{4}\right)\sqrt{7}\right)\times\left(\dfrac{33}{4}-\left(\dfrac{25}{2}-\left(\left(-15\right)\sqrt{7}\right)-\left(\left(-\dfrac{50}{9}\right)\sqrt{7}\right)\right)\right)\\
&=&\left(\left(\dfrac{7201}{168}\right)\sqrt{7}-\dfrac{133}{4}\right)\left(-\dfrac{17}{4}+\left(-\dfrac{185}{9}\right)\sqrt{7}\right)\\
&=&\left(\dfrac{1010629}{2016}\right)\sqrt{7}+\left(-\dfrac{1332185}{1512}\right)\sqrt{49}+\dfrac{2261}{16}\\
&=&\left(\dfrac{1010629}{2016}\right)\sqrt{7}-\dfrac{2603323}{432}\\
\end{eqnarray*}