L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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La correction se trouve en bas de page.
Exercice
Soit \( X=\left(\dfrac{17}{2}-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(-\dfrac{5}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\left(\left(\dfrac{35}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{8}\right)-\dfrac{43}{8}\right)\) et \( Y=\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(\dfrac{33}{8}\right)\sqrt{18}+\left(8\right)\sqrt{8}+\left(\dfrac{27}{2}\right)\sqrt{18}+\left(-\dfrac{59}{3}\right)\sqrt{18}\) . 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{17}{2}-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(-\dfrac{5}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\left(\left(\dfrac{35}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{8}\right)-\dfrac{43}{8}\right)\right)+\left(\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(\dfrac{33}{8}\right)\sqrt{18}+\left(8\right)\sqrt{8}+\left(\dfrac{27}{2}\right)\sqrt{18}+\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{17}{2}-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-5\right)\sqrt{2}-\dfrac{10}{7}\right)-\left(-37+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\dfrac{70}{3}-\left(\left(\dfrac{70}{9}\right)\sqrt{2}\right)-\dfrac{43}{8}\right)\right)+\left(\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(\dfrac{99}{8}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(\dfrac{81}{2}\right)\sqrt{2}+\left(-59\right)\sqrt{2}\right)\\
&=&\left(\dfrac{17}{2}-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-5\right)\sqrt{2}-\dfrac{10}{7}\right)-\left(-37+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\dfrac{70}{3}-\left(\left(\dfrac{70}{9}\right)\sqrt{2}\right)-\dfrac{43}{8}\right)+\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(\dfrac{99}{8}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(\dfrac{81}{2}\right)\sqrt{2}+\left(-59\right)\sqrt{2}\\
&=&\dfrac{9851}{168}+\left(\dfrac{7115}{504}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\dfrac{17}{2}-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(-\dfrac{5}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\left(\left(\dfrac{35}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{8}\right)-\dfrac{43}{8}\right)\right)-\left(\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(\dfrac{33}{8}\right)\sqrt{18}+\left(8\right)\sqrt{8}+\left(\dfrac{27}{2}\right)\sqrt{18}+\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{17}{2}-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-5\right)\sqrt{2}-\dfrac{10}{7}\right)-\left(-37+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\dfrac{70}{3}-\left(\left(\dfrac{70}{9}\right)\sqrt{2}\right)-\dfrac{43}{8}\right)\right)-\left(\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(\dfrac{99}{8}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(\dfrac{81}{2}\right)\sqrt{2}+\left(-59\right)\sqrt{2}\right)\\
&=&\left(\dfrac{9851}{168}+\left(\dfrac{1741}{252}\right)\sqrt{2}\right)-\left(\left(\dfrac{173}{24}\right)\sqrt{2}\right)\\
&=&\dfrac{9851}{168}+\left(\dfrac{1741}{252}\right)\sqrt{2}+\left(-\dfrac{173}{24}\right)\sqrt{2}\\
&=&\dfrac{9851}{168}+\left(-\dfrac{151}{504}\right)\sqrt{2}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\dfrac{17}{2}-\left(\left(-\dfrac{81}{7}\right)\sqrt{18}\right)-\left(\left(\dfrac{32}{3}\right)\sqrt{50}\right)\right)-\left(\left(-\dfrac{3}{8}\right)\sqrt{8}+\left(-6\right)\sqrt{8}+\left(-1\right)\sqrt{50}+\left(-\dfrac{5}{7}\right)\sqrt{4}\right)-\left(\left(-\dfrac{37}{2}\right)\sqrt{4}+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\left(\left(\dfrac{35}{3}\right)\sqrt{4}\right)-\left(\left(\dfrac{35}{9}\right)\sqrt{8}\right)-\dfrac{43}{8}\right)\right)\times\left(\left(-\dfrac{4}{3}\right)\sqrt{8}+\left(\dfrac{33}{8}\right)\sqrt{18}+\left(8\right)\sqrt{8}+\left(\dfrac{27}{2}\right)\sqrt{18}+\left(-\dfrac{59}{3}\right)\sqrt{18}\right)\\
&=&\left(\left(\dfrac{17}{2}-\left(\left(-\dfrac{243}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)\right)-\left(\left(-\dfrac{3}{4}\right)\sqrt{2}+\left(-12\right)\sqrt{2}+\left(-5\right)\sqrt{2}-\dfrac{10}{7}\right)-\left(-37+\dfrac{19}{2}-2\right)-\left(\dfrac{19}{2}-\dfrac{70}{3}-\left(\left(\dfrac{70}{9}\right)\sqrt{2}\right)-\dfrac{43}{8}\right)\right)\times\left(\left(-\dfrac{8}{3}\right)\sqrt{2}+\left(\dfrac{99}{8}\right)\sqrt{2}+\left(16\right)\sqrt{2}+\left(\dfrac{81}{2}\right)\sqrt{2}+\left(-59\right)\sqrt{2}\right)\\
&=&\left(\dfrac{9851}{168}+\left(\dfrac{1741}{252}\right)\sqrt{2}\right)\left(\left(\dfrac{173}{24}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{1704223}{4032}\right)\sqrt{2}+\left(\dfrac{301193}{6048}\right)\sqrt{4}\\
&=&\left(\dfrac{1704223}{4032}\right)\sqrt{2}+\dfrac{301193}{3024}\\
\end{eqnarray*}