L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(-\dfrac{52}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{40}{7}\right)\sqrt{18}\right)-\dfrac{33}{2}+\left(\left(\dfrac{13}{4}\right)\sqrt{18}\right)-\dfrac{5}{6}\) et \( Y=\left(\dfrac{46}{3}-\left(\left(\dfrac{39}{8}\right)\sqrt{4}\right)\right)-\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{8}\right)+\dfrac{71}{4}\right)-\left(\left(-\dfrac{27}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{4}\right)\right)\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{40}{7}\right)\sqrt{18}\right)-\dfrac{33}{2}+\left(\left(\dfrac{13}{4}\right)\sqrt{18}\right)-\dfrac{5}{6}\right)+\left(\left(\dfrac{46}{3}-\left(\left(\dfrac{39}{8}\right)\sqrt{4}\right)\right)-\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{8}\right)+\dfrac{71}{4}\right)-\left(\left(-\dfrac{27}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{4}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{120}{7}\right)\sqrt{2}\right)-\dfrac{33}{2}+\left(\left(\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{5}{6}\right)+\left(\left(\dfrac{46}{3}-\dfrac{39}{4}\right)-\left(\left(\left(-67\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\dfrac{71}{4}\right)+\dfrac{54}{7}-\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)-\dfrac{29}{2}\right)\right)\\
&=&\left(\left(-\dfrac{104}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{120}{7}\right)\sqrt{2}\right)-\dfrac{33}{2}+\left(\left(\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{5}{6}+\left(\dfrac{46}{3}-\dfrac{39}{4}\right)-\left(\left(\left(-67\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\dfrac{71}{4}\right)+\dfrac{54}{7}-\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)-\dfrac{29}{2}\right)\\
&=&\left(\dfrac{27779}{420}\right)\sqrt{2}-\dfrac{51}{7}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{40}{7}\right)\sqrt{18}\right)-\dfrac{33}{2}+\left(\left(\dfrac{13}{4}\right)\sqrt{18}\right)-\dfrac{5}{6}\right)-\left(\left(\dfrac{46}{3}-\left(\left(\dfrac{39}{8}\right)\sqrt{4}\right)\right)-\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{8}\right)+\dfrac{71}{4}\right)-\left(\left(-\dfrac{27}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{4}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{120}{7}\right)\sqrt{2}\right)-\dfrac{33}{2}+\left(\left(\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{5}{6}\right)-\left(\left(\dfrac{46}{3}-\dfrac{39}{4}\right)-\left(\left(\left(-67\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\dfrac{71}{4}\right)+\dfrac{54}{7}-\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)-\dfrac{29}{2}\right)\right)\\
&=&\left(\left(-\dfrac{3947}{140}\right)\sqrt{2}-\dfrac{52}{3}\right)-\left(\dfrac{211}{21}+\left(\dfrac{283}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{3947}{140}\right)\sqrt{2}-\dfrac{52}{3}+-\dfrac{211}{21}+\left(-\dfrac{283}{3}\right)\sqrt{2}\\
&=&\left(-\dfrac{51461}{420}\right)\sqrt{2}-\dfrac{575}{21}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{52}{5}\right)\sqrt{8}\right)-\left(\left(\dfrac{40}{7}\right)\sqrt{18}\right)-\dfrac{33}{2}+\left(\left(\dfrac{13}{4}\right)\sqrt{18}\right)-\dfrac{5}{6}\right)\times\left(\left(\dfrac{46}{3}-\left(\left(\dfrac{39}{8}\right)\sqrt{4}\right)\right)-\left(\left(\left(-\dfrac{67}{3}\right)\sqrt{18}\right)-\left(\left(7\right)\sqrt{8}\right)+\dfrac{71}{4}\right)-\left(\left(-\dfrac{27}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{40}{9}\right)\sqrt{18}\right)-\left(\left(\dfrac{29}{4}\right)\sqrt{4}\right)\right)\right)\\
&=&\left(\left(\left(-\dfrac{104}{5}\right)\sqrt{2}\right)-\left(\left(\dfrac{120}{7}\right)\sqrt{2}\right)-\dfrac{33}{2}+\left(\left(\dfrac{39}{4}\right)\sqrt{2}\right)-\dfrac{5}{6}\right)\times\left(\left(\dfrac{46}{3}-\dfrac{39}{4}\right)-\left(\left(\left(-67\right)\sqrt{2}\right)-\left(\left(14\right)\sqrt{2}\right)+\dfrac{71}{4}\right)+\dfrac{54}{7}-\left(\left(\left(-\dfrac{40}{3}\right)\sqrt{2}\right)-\dfrac{29}{2}\right)\right)\\
&=&\left(\left(-\dfrac{3947}{140}\right)\sqrt{2}-\dfrac{52}{3}\right)\left(\dfrac{211}{21}+\left(\dfrac{283}{3}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{16920131}{8820}\right)\sqrt{2}+\left(-\dfrac{1117001}{420}\right)\sqrt{4}-\dfrac{10972}{63}\\
&=&\left(-\dfrac{16920131}{8820}\right)\sqrt{2}-\dfrac{494389}{90}\\
\end{eqnarray*}