L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\dfrac{71}{7}-\left(\left(-\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{50}\right)+\left(\left(-\dfrac{3}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\dfrac{17}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{8}\right)\) et \( Y=\left(\left(-5\right)\sqrt{18}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\dfrac{67}{8}+\left(\left(\dfrac{32}{3}\right)\sqrt{8}\right)-\dfrac{13}{8}+\left(\dfrac{63}{2}\right)\sqrt{4}\) . Calculer et simplifier \( X+Y\) , \( X-Y\) et \( X\times Y\) .
Cliquer ici pour afficher la solution
Exercice
\begin{eqnarray*}
X+Y
&=&\left(\dfrac{71}{7}-\left(\left(-\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{50}\right)+\left(\left(-\dfrac{3}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\dfrac{17}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{8}\right)\right)+\left(\left(\left(-5\right)\sqrt{18}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\dfrac{67}{8}+\left(\left(\dfrac{32}{3}\right)\sqrt{8}\right)-\dfrac{13}{8}+\left(\dfrac{63}{2}\right)\sqrt{4}\right)\\
&=&\left(\dfrac{71}{7}-\left(\left(-\dfrac{135}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\left(-3\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\dfrac{17}{5}+27-\left(\left(59\right)\sqrt{2}\right)\right)+\left(\left(\left(-15\right)\sqrt{2}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)-\dfrac{11}{2}+\dfrac{67}{8}+\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)-\dfrac{13}{8}+63\right)\\
&=&\dfrac{71}{7}-\left(\left(-\dfrac{135}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\left(-3\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\dfrac{17}{5}+27-\left(\left(59\right)\sqrt{2}\right)+\left(\left(-15\right)\sqrt{2}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)-\dfrac{11}{2}+\dfrac{67}{8}+\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)-\dfrac{13}{8}+63\\
&=&\dfrac{2037}{20}+\left(-\dfrac{4357}{72}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\dfrac{71}{7}-\left(\left(-\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{50}\right)+\left(\left(-\dfrac{3}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\dfrac{17}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{8}\right)\right)-\left(\left(\left(-5\right)\sqrt{18}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\dfrac{67}{8}+\left(\left(\dfrac{32}{3}\right)\sqrt{8}\right)-\dfrac{13}{8}+\left(\dfrac{63}{2}\right)\sqrt{4}\right)\\
&=&\left(\dfrac{71}{7}-\left(\left(-\dfrac{135}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\left(-3\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\dfrac{17}{5}+27-\left(\left(59\right)\sqrt{2}\right)\right)-\left(\left(\left(-15\right)\sqrt{2}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)-\dfrac{11}{2}+\dfrac{67}{8}+\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)-\dfrac{13}{8}+63\right)\\
&=&\left(\dfrac{1181}{35}+\left(-\dfrac{4921}{72}\right)\sqrt{2}\right)-\left(\left(\dfrac{47}{6}\right)\sqrt{2}+\dfrac{1907}{28}\right)\\
&=&\dfrac{1181}{35}+\left(-\dfrac{4921}{72}\right)\sqrt{2}+\left(-\dfrac{47}{6}\right)\sqrt{2}-\dfrac{1907}{28}\\
&=&-\dfrac{4811}{140}+\left(-\dfrac{5485}{72}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\dfrac{71}{7}-\left(\left(-\dfrac{45}{8}\right)\sqrt{18}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\left(\left(5\right)\sqrt{50}\right)+\left(\left(-\dfrac{3}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{4}{9}\right)\sqrt{8}\right)-\dfrac{17}{5}-\left(\left(-\dfrac{27}{2}\right)\sqrt{4}\right)-\left(\left(\dfrac{59}{2}\right)\sqrt{8}\right)\right)\times\left(\left(\left(-5\right)\sqrt{18}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(-\dfrac{1}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{11}{4}\right)\sqrt{4}\right)+\dfrac{67}{8}+\left(\left(\dfrac{32}{3}\right)\sqrt{8}\right)-\dfrac{13}{8}+\left(\dfrac{63}{2}\right)\sqrt{4}\right)\\
&=&\left(\dfrac{71}{7}-\left(\left(-\dfrac{135}{8}\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}\right)+\left(\left(-3\right)\sqrt{2}\right)-\left(\left(-\dfrac{8}{9}\right)\sqrt{2}\right)-\dfrac{17}{5}+27-\left(\left(59\right)\sqrt{2}\right)\right)\times\left(\left(\left(-15\right)\sqrt{2}\right)+\dfrac{27}{7}+\left(\left(0\right)\sqrt{2}\right)-\left(\left(-\dfrac{3}{2}\right)\sqrt{2}\right)-\dfrac{11}{2}+\dfrac{67}{8}+\left(\left(\dfrac{64}{3}\right)\sqrt{2}\right)-\dfrac{13}{8}+63\right)\\
&=&\left(\dfrac{1181}{35}+\left(-\dfrac{4921}{72}\right)\sqrt{2}\right)\left(\left(\dfrac{47}{6}\right)\sqrt{2}+\dfrac{1907}{28}\right)\\
&=&\left(-\dfrac{44257399}{10080}\right)\sqrt{2}+\dfrac{2252167}{980}+\left(-\dfrac{231287}{432}\right)\sqrt{4}\\
&=&\left(-\dfrac{44257399}{10080}\right)\sqrt{2}+\dfrac{64951703}{52920}\\
\end{eqnarray*}