L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{59}{7}\right)\sqrt{18}\) et \( Y=\left(\left(\dfrac{53}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{8}+\left(-\dfrac{40}{9}\right)\sqrt{8}+\left(-\dfrac{41}{3}\right)\sqrt{8}+\left(-\dfrac{47}{4}\right)\sqrt{50}+\dfrac{10}{7}\right)-\left(-7-3+\left(9\right)\sqrt{4}+\left(2\right)\sqrt{4}\right)+\dfrac{16}{5}\) . 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{59}{7}\right)\sqrt{18}\right)+\left(\left(\left(\dfrac{53}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{8}+\left(-\dfrac{40}{9}\right)\sqrt{8}+\left(-\dfrac{41}{3}\right)\sqrt{8}+\left(-\dfrac{47}{4}\right)\sqrt{50}+\dfrac{10}{7}\right)-\left(-7-3+\left(9\right)\sqrt{4}+\left(2\right)\sqrt{4}\right)+\dfrac{16}{5}\right)\\
&=&\left(\left(\dfrac{177}{7}\right)\sqrt{2}\right)+\left(\left(\left(\dfrac{106}{9}\right)\sqrt{2}\right)+\dfrac{11}{2}-\left(\left(\dfrac{106}{9}\right)\sqrt{2}+\left(-\dfrac{80}{9}\right)\sqrt{2}+\left(-\dfrac{82}{3}\right)\sqrt{2}+\left(-\dfrac{235}{4}\right)\sqrt{2}+\dfrac{10}{7}\right)-\left(-7-3+18+4\right)+\dfrac{16}{5}\right)\\
&=&\left(\dfrac{177}{7}\right)\sqrt{2}+\left(\left(\dfrac{106}{9}\right)\sqrt{2}\right)+\dfrac{11}{2}-\left(\left(\dfrac{106}{9}\right)\sqrt{2}+\left(-\dfrac{80}{9}\right)\sqrt{2}+\left(-\dfrac{82}{3}\right)\sqrt{2}+\left(-\dfrac{235}{4}\right)\sqrt{2}+\dfrac{10}{7}\right)-\left(-7-3+18+4\right)+\dfrac{16}{5}\\
&=&\left(\dfrac{30305}{252}\right)\sqrt{2}-\dfrac{331}{70}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{59}{7}\right)\sqrt{18}\right)-\left(\left(\left(\dfrac{53}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{8}+\left(-\dfrac{40}{9}\right)\sqrt{8}+\left(-\dfrac{41}{3}\right)\sqrt{8}+\left(-\dfrac{47}{4}\right)\sqrt{50}+\dfrac{10}{7}\right)-\left(-7-3+\left(9\right)\sqrt{4}+\left(2\right)\sqrt{4}\right)+\dfrac{16}{5}\right)\\
&=&\left(\left(\dfrac{177}{7}\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{106}{9}\right)\sqrt{2}\right)+\dfrac{11}{2}-\left(\left(\dfrac{106}{9}\right)\sqrt{2}+\left(-\dfrac{80}{9}\right)\sqrt{2}+\left(-\dfrac{82}{3}\right)\sqrt{2}+\left(-\dfrac{235}{4}\right)\sqrt{2}+\dfrac{10}{7}\right)-\left(-7-3+18+4\right)+\dfrac{16}{5}\right)\\
&=&\left(\left(\dfrac{177}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{3419}{36}\right)\sqrt{2}-\dfrac{331}{70}\right)\\
&=&\left(\dfrac{177}{7}\right)\sqrt{2}+\left(-\dfrac{3419}{36}\right)\sqrt{2}+\dfrac{331}{70}\\
&=&\left(-\dfrac{17561}{252}\right)\sqrt{2}+\dfrac{331}{70}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{59}{7}\right)\sqrt{18}\right)\times\left(\left(\left(\dfrac{53}{9}\right)\sqrt{8}\right)-\left(\left(-\dfrac{11}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{53}{9}\right)\sqrt{8}+\left(-\dfrac{40}{9}\right)\sqrt{8}+\left(-\dfrac{41}{3}\right)\sqrt{8}+\left(-\dfrac{47}{4}\right)\sqrt{50}+\dfrac{10}{7}\right)-\left(-7-3+\left(9\right)\sqrt{4}+\left(2\right)\sqrt{4}\right)+\dfrac{16}{5}\right)\\
&=&\left(\left(\dfrac{177}{7}\right)\sqrt{2}\right)\times\left(\left(\left(\dfrac{106}{9}\right)\sqrt{2}\right)+\dfrac{11}{2}-\left(\left(\dfrac{106}{9}\right)\sqrt{2}+\left(-\dfrac{80}{9}\right)\sqrt{2}+\left(-\dfrac{82}{3}\right)\sqrt{2}+\left(-\dfrac{235}{4}\right)\sqrt{2}+\dfrac{10}{7}\right)-\left(-7-3+18+4\right)+\dfrac{16}{5}\right)\\
&=&\left(\left(\dfrac{177}{7}\right)\sqrt{2}\right)\left(\left(\dfrac{3419}{36}\right)\sqrt{2}-\dfrac{331}{70}\right)\\
&=&\left(\dfrac{201721}{84}\right)\sqrt{4}+\left(-\dfrac{58587}{490}\right)\sqrt{2}\\
&=&\dfrac{201721}{42}+\left(-\dfrac{58587}{490}\right)\sqrt{2}\\
\end{eqnarray*}