L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\left(\left(\dfrac{13}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{79}{6}\right)\sqrt{18}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{6}\right)\sqrt{8}\right)\) et \( Y=\left(\left(\dfrac{23}{2}\right)\sqrt{8}+\dfrac{27}{5}\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{4}+\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{8}+\left(\dfrac{23}{2}\right)\sqrt{50}\right)\) . 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(\left(\dfrac{13}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{79}{6}\right)\sqrt{18}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{6}\right)\sqrt{8}\right)\right)+\left(\left(\left(\dfrac{23}{2}\right)\sqrt{8}+\dfrac{27}{5}\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{4}+\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{8}+\left(\dfrac{23}{2}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{26}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)\right)+\left(\left(\left(23\right)\sqrt{2}+\dfrac{27}{5}\right)-\left(\dfrac{5}{4}+\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{2}+\left(\dfrac{115}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\left(\dfrac{26}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)+\left(\left(23\right)\sqrt{2}+\dfrac{27}{5}\right)-\left(\dfrac{5}{4}+\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{2}+\left(\dfrac{115}{2}\right)\sqrt{2}\right)\\
&=&\left(-\dfrac{8177}{84}\right)\sqrt{2}+\dfrac{89}{60}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(\left(\dfrac{13}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{79}{6}\right)\sqrt{18}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{6}\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{23}{2}\right)\sqrt{8}+\dfrac{27}{5}\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{4}+\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{8}+\left(\dfrac{23}{2}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{26}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)\right)-\left(\left(\left(23\right)\sqrt{2}+\dfrac{27}{5}\right)-\left(\dfrac{5}{4}+\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{2}+\left(\dfrac{115}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{259}{6}\right)\sqrt{2}-\dfrac{8}{3}\right)-\left(\left(-\dfrac{1517}{28}\right)\sqrt{2}+\dfrac{83}{20}\right)\\
&=&\left(-\dfrac{259}{6}\right)\sqrt{2}-\dfrac{8}{3}+\left(\dfrac{1517}{28}\right)\sqrt{2}-\dfrac{83}{20}\\
&=&\left(\dfrac{925}{84}\right)\sqrt{2}-\dfrac{409}{60}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(\left(\dfrac{13}{3}\right)\sqrt{8}\right)-\left(\left(\dfrac{79}{6}\right)\sqrt{18}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{6}\right)\sqrt{8}\right)\right)\times\left(\left(\left(\dfrac{23}{2}\right)\sqrt{8}+\dfrac{27}{5}\right)-\left(\left(\dfrac{5}{8}\right)\sqrt{4}+\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(\dfrac{75}{7}\right)\sqrt{8}+\left(\dfrac{23}{2}\right)\sqrt{50}\right)\right)\\
&=&\left(\left(\left(\left(\dfrac{26}{3}\right)\sqrt{2}\right)-\left(\left(\dfrac{79}{2}\right)\sqrt{2}\right)-\dfrac{8}{3}\right)-\left(\left(\dfrac{37}{3}\right)\sqrt{2}\right)\right)\times\left(\left(\left(23\right)\sqrt{2}+\dfrac{27}{5}\right)-\left(\dfrac{5}{4}+\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{150}{7}\right)\sqrt{2}+\left(\dfrac{115}{2}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{259}{6}\right)\sqrt{2}-\dfrac{8}{3}\right)\left(\left(-\dfrac{1517}{28}\right)\sqrt{2}+\dfrac{83}{20}\right)\\
&=&\left(\dfrac{56129}{24}\right)\sqrt{4}+\left(-\dfrac{29119}{840}\right)\sqrt{2}-\dfrac{166}{15}\\
&=&\dfrac{93327}{20}+\left(-\dfrac{29119}{840}\right)\sqrt{2}\\
\end{eqnarray*}