L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
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Exercice
Soit \( X=\left(\left(-\dfrac{43}{9}\right)\sqrt{50}+\left(2\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{18}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{10}{3}\right)\sqrt{50}+\left(-9\right)\sqrt{50}\right)\) et \( Y=\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{61}{3}\right)\sqrt{8}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{8}\right)\) . 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{43}{9}\right)\sqrt{50}+\left(2\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{18}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{10}{3}\right)\sqrt{50}+\left(-9\right)\sqrt{50}\right)\right)+\left(\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{61}{3}\right)\sqrt{8}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-\dfrac{215}{9}\right)\sqrt{2}+4\right)-\left(\left(\left(-\dfrac{60}{7}\right)\sqrt{2}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(20\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{50}{3}\right)\sqrt{2}+\left(-45\right)\sqrt{2}\right)\right)+\left(\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{122}{3}\right)\sqrt{2}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(-\dfrac{215}{9}\right)\sqrt{2}+4\right)-\left(\left(\left(-\dfrac{60}{7}\right)\sqrt{2}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(20\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{50}{3}\right)\sqrt{2}+\left(-45\right)\sqrt{2}\right)+\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{122}{3}\right)\sqrt{2}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{2}\right)\\
&=&\left(\dfrac{9613}{126}\right)\sqrt{2}-\dfrac{43}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X-Y
&=&\left(\left(\left(-\dfrac{43}{9}\right)\sqrt{50}+\left(2\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{18}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{10}{3}\right)\sqrt{50}+\left(-9\right)\sqrt{50}\right)\right)-\left(\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{61}{3}\right)\sqrt{8}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-\dfrac{215}{9}\right)\sqrt{2}+4\right)-\left(\left(\left(-\dfrac{60}{7}\right)\sqrt{2}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(20\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{50}{3}\right)\sqrt{2}+\left(-45\right)\sqrt{2}\right)\right)-\left(\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{122}{3}\right)\sqrt{2}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{946}{63}\right)\sqrt{2}-\dfrac{59}{2}\right)-\left(\left(\dfrac{1103}{18}\right)\sqrt{2}+\dfrac{91}{6}\right)\\
&=&\left(\dfrac{946}{63}\right)\sqrt{2}-\dfrac{59}{2}+\left(-\dfrac{1103}{18}\right)\sqrt{2}-\dfrac{91}{6}\\
&=&\left(-\dfrac{1943}{42}\right)\sqrt{2}-\dfrac{134}{3}\\
\end{eqnarray*}
\begin{eqnarray*}
X\times Y
&=&\left(\left(\left(-\dfrac{43}{9}\right)\sqrt{50}+\left(2\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{20}{7}\right)\sqrt{18}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(4\right)\sqrt{50}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{18}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{10}{3}\right)\sqrt{50}+\left(-9\right)\sqrt{50}\right)\right)\times\left(\left(\left(\dfrac{56}{9}\right)\sqrt{8}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{61}{3}\right)\sqrt{8}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{4}\right)\sqrt{8}\right)-\left(\left(-\dfrac{35}{6}\right)\sqrt{8}\right)\right)\\
&=&\left(\left(\left(-\dfrac{215}{9}\right)\sqrt{2}+4\right)-\left(\left(\left(-\dfrac{60}{7}\right)\sqrt{2}\right)-7+\dfrac{7}{2}\right)-\left(\left(\left(20\right)\sqrt{2}\right)-\left(\left(22\right)\sqrt{2}\right)+\dfrac{79}{2}\right)+\dfrac{5}{2}-\left(\left(\dfrac{50}{3}\right)\sqrt{2}+\left(-45\right)\sqrt{2}\right)\right)\times\left(\left(\left(\dfrac{112}{9}\right)\sqrt{2}\right)-\left(-\dfrac{5}{3}+\left(-\dfrac{122}{3}\right)\sqrt{2}\right)-\left(-\dfrac{27}{2}+\left(\dfrac{7}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{35}{3}\right)\sqrt{2}\right)\right)\\
&=&\left(\left(\dfrac{946}{63}\right)\sqrt{2}-\dfrac{59}{2}\right)\left(\left(\dfrac{1103}{18}\right)\sqrt{2}+\dfrac{91}{6}\right)\\
&=&\left(\dfrac{521719}{567}\right)\sqrt{4}+\left(-\dfrac{170635}{108}\right)\sqrt{2}-\dfrac{5369}{12}\\
&=&\dfrac{3159011}{2268}+\left(-\dfrac{170635}{108}\right)\sqrt{2}\\
\end{eqnarray*}