L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(\dfrac{13}{6}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{81}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\dfrac{14}{5}+\left(\dfrac{32}{5}\right)\sqrt{4}+\left(\dfrac{27}{4}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(-\dfrac{47}{8}\right)\sqrt{4}\right)-\dfrac{35}{3}\) et \( Y=\left(-\dfrac{41}{6}\right)\sqrt{4}+\left(-\dfrac{18}{5}\right)\sqrt{8}\) . 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{13}{6}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{81}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\dfrac{14}{5}+\left(\dfrac{32}{5}\right)\sqrt{4}+\left(\dfrac{27}{4}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(-\dfrac{47}{8}\right)\sqrt{4}\right)-\dfrac{35}{3}\right)+\left(\left(-\dfrac{41}{6}\right)\sqrt{4}+\left(-\dfrac{18}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{65}{6}\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{243}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)-\left(\left(5\right)\sqrt{2}\right)\right)-\left(\dfrac{14}{5}+\dfrac{64}{5}+\dfrac{27}{2}+\left(5\right)\sqrt{2}-\dfrac{47}{4}\right)-\dfrac{35}{3}\right)+\left(-\dfrac{41}{3}+\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(\dfrac{65}{6}\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{243}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)-\left(\left(5\right)\sqrt{2}\right)\right)-\left(\dfrac{14}{5}+\dfrac{64}{5}+\dfrac{27}{2}+\left(5\right)\sqrt{2}-\dfrac{47}{4}\right)-\dfrac{35}{3}-\dfrac{41}{3}+\left(-\dfrac{36}{5}\right)\sqrt{2}\\
&=&\left(-\dfrac{709}{20}\right)\sqrt{2}-\dfrac{2561}{60}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\dfrac{13}{6}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{81}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\dfrac{14}{5}+\left(\dfrac{32}{5}\right)\sqrt{4}+\left(\dfrac{27}{4}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(-\dfrac{47}{8}\right)\sqrt{4}\right)-\dfrac{35}{3}\right)-\left(\left(-\dfrac{41}{6}\right)\sqrt{4}+\left(-\dfrac{18}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{65}{6}\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{243}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)-\left(\left(5\right)\sqrt{2}\right)\right)-\left(\dfrac{14}{5}+\dfrac{64}{5}+\dfrac{27}{2}+\left(5\right)\sqrt{2}-\dfrac{47}{4}\right)-\dfrac{35}{3}\right)-\left(-\dfrac{41}{3}+\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{113}{4}\right)\sqrt{2}-\dfrac{1741}{60}\right)-\left(-\dfrac{41}{3}+\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{113}{4}\right)\sqrt{2}-\dfrac{1741}{60}+\dfrac{41}{3}+\left(\dfrac{36}{5}\right)\sqrt{2}\\
&=&\left(-\dfrac{421}{20}\right)\sqrt{2}-\dfrac{307}{20}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\dfrac{13}{6}\right)\sqrt{50}\right)-\left(\left(\left(\dfrac{81}{4}\right)\sqrt{18}\right)-\left(\left(\dfrac{13}{3}\right)\sqrt{50}\right)-\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\dfrac{14}{5}+\left(\dfrac{32}{5}\right)\sqrt{4}+\left(\dfrac{27}{4}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(-\dfrac{47}{8}\right)\sqrt{4}\right)-\dfrac{35}{3}\right)\times\left(\left(-\dfrac{41}{6}\right)\sqrt{4}+\left(-\dfrac{18}{5}\right)\sqrt{8}\right)\\
&=&\left(\left(\left(\dfrac{65}{6}\right)\sqrt{2}\right)-\left(\left(\left(\dfrac{243}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{65}{3}\right)\sqrt{2}\right)-\left(\left(5\right)\sqrt{2}\right)\right)-\left(\dfrac{14}{5}+\dfrac{64}{5}+\dfrac{27}{2}+\left(5\right)\sqrt{2}-\dfrac{47}{4}\right)-\dfrac{35}{3}\right)\times\left(-\dfrac{41}{3}+\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\left(-\dfrac{113}{4}\right)\sqrt{2}-\dfrac{1741}{60}\right)\left(-\dfrac{41}{3}+\left(-\dfrac{36}{5}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{178501}{300}\right)\sqrt{2}+\left(\dfrac{1017}{5}\right)\sqrt{4}+\dfrac{71381}{180}\\
&=&\left(\dfrac{178501}{300}\right)\sqrt{2}+\dfrac{28921}{36}\\
\end{eqnarray*}