Ataraxy

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Exercice

L'exercice suivant est automatiquement et aléatoirement généré par ataraXy.
Si vous regénérez la page (F5) les valeurs seront changées.
La correction se trouve en bas de page.

Exercice

Soit $ X=\left(-\dfrac{17}{6}\right)\sqrt{8}+\dfrac{15}{2}-\left(\left(4\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{6}\right)\sqrt{8}\right)-\left(\left(2\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-9-\left(\left(\dfrac{47}{2}\right)\sqrt{8}\right)+\dfrac{48}{7}+\dfrac{3}{7}$ et $ Y=\left(\left(\dfrac{5}{4}\right)\sqrt{8}+\left(-\dfrac{45}{2}\right)\sqrt{18}+\left(\dfrac{37}{5}\right)\sqrt{8}-\dfrac{53}{4}+\left(-\dfrac{19}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(\dfrac{19}{8}\right)\sqrt{18}+\left(\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{50}{3}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{80}{3}\right)\sqrt{8}\right)-\dfrac{23}{5}\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(-\dfrac{17}{6}\right)\sqrt{8}+\dfrac{15}{2}-\left(\left(4\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{6}\right)\sqrt{8}\right)-\left(\left(2\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-9-\left(\left(\dfrac{47}{2}\right)\sqrt{8}\right)+\dfrac{48}{7}+\dfrac{3}{7}\right)+\left(\left(\left(\dfrac{5}{4}\right)\sqrt{8}+\left(-\dfrac{45}{2}\right)\sqrt{18}+\left(\dfrac{37}{5}\right)\sqrt{8}-\dfrac{53}{4}+\left(-\dfrac{19}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(\dfrac{19}{8}\right)\sqrt{18}+\left(\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{50}{3}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{80}{3}\right)\sqrt{8}\right)-\dfrac{23}{5}\right)\right)\\ &=&\left(\left(-\dfrac{17}{3}\right)\sqrt{2}+\dfrac{15}{2}-8-\left(\left(-\dfrac{17}{3}\right)\sqrt{2}\right)-\left(\left(4\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)-9-\left(\left(47\right)\sqrt{2}\right)+\dfrac{48}{7}+\dfrac{3}{7}\right)+\left(\left(\left(\dfrac{5}{2}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\dfrac{74}{5}\right)\sqrt{2}-\dfrac{53}{4}+\left(-\dfrac{57}{2}\right)\sqrt{2}\right)-\left(\dfrac{144}{7}+\left(\dfrac{57}{8}\right)\sqrt{2}+\dfrac{7}{2}\right)-\left(33+\dfrac{100}{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)-\dfrac{23}{5}\right)\right)\\ &=&\left(-\dfrac{17}{3}\right)\sqrt{2}+\dfrac{15}{2}-8-\left(\left(-\dfrac{17}{3}\right)\sqrt{2}\right)-\left(\left(4\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)-9-\left(\left(47\right)\sqrt{2}\right)+\dfrac{48}{7}+\dfrac{3}{7}+\left(\left(\dfrac{5}{2}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\dfrac{74}{5}\right)\sqrt{2}-\dfrac{53}{4}+\left(-\dfrac{57}{2}\right)\sqrt{2}\right)-\left(\dfrac{144}{7}+\left(\dfrac{57}{8}\right)\sqrt{2}+\dfrac{7}{2}\right)-\left(33+\dfrac{100}{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)-\dfrac{23}{5}\right)\\ &=&\left(-\dfrac{26059}{120}\right)\sqrt{2}-\dfrac{42533}{420}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(-\dfrac{17}{6}\right)\sqrt{8}+\dfrac{15}{2}-\left(\left(4\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{6}\right)\sqrt{8}\right)-\left(\left(2\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-9-\left(\left(\dfrac{47}{2}\right)\sqrt{8}\right)+\dfrac{48}{7}+\dfrac{3}{7}\right)-\left(\left(\left(\dfrac{5}{4}\right)\sqrt{8}+\left(-\dfrac{45}{2}\right)\sqrt{18}+\left(\dfrac{37}{5}\right)\sqrt{8}-\dfrac{53}{4}+\left(-\dfrac{19}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(\dfrac{19}{8}\right)\sqrt{18}+\left(\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{50}{3}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{80}{3}\right)\sqrt{8}\right)-\dfrac{23}{5}\right)\right)\\ &=&\left(\left(-\dfrac{17}{3}\right)\sqrt{2}+\dfrac{15}{2}-8-\left(\left(-\dfrac{17}{3}\right)\sqrt{2}\right)-\left(\left(4\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)-9-\left(\left(47\right)\sqrt{2}\right)+\dfrac{48}{7}+\dfrac{3}{7}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\dfrac{74}{5}\right)\sqrt{2}-\dfrac{53}{4}+\left(-\dfrac{57}{2}\right)\sqrt{2}\right)-\left(\dfrac{144}{7}+\left(\dfrac{57}{8}\right)\sqrt{2}+\dfrac{7}{2}\right)-\left(33+\dfrac{100}{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)-\dfrac{23}{5}\right)\right)\\ &=&\left(\left(-78\right)\sqrt{2}-\dfrac{31}{14}\right)-\left(\left(-\dfrac{16699}{120}\right)\sqrt{2}-\dfrac{41603}{420}\right)\\ &=&\left(-78\right)\sqrt{2}-\dfrac{31}{14}+\left(\dfrac{16699}{120}\right)\sqrt{2}+\dfrac{41603}{420}\\ &=&\left(\dfrac{7339}{120}\right)\sqrt{2}+\dfrac{40673}{420}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(-\dfrac{17}{6}\right)\sqrt{8}+\dfrac{15}{2}-\left(\left(4\right)\sqrt{4}\right)-\left(\left(-\dfrac{17}{6}\right)\sqrt{8}\right)-\left(\left(2\right)\sqrt{8}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{8}\right)-9-\left(\left(\dfrac{47}{2}\right)\sqrt{8}\right)+\dfrac{48}{7}+\dfrac{3}{7}\right)\times\left(\left(\left(\dfrac{5}{4}\right)\sqrt{8}+\left(-\dfrac{45}{2}\right)\sqrt{18}+\left(\dfrac{37}{5}\right)\sqrt{8}-\dfrac{53}{4}+\left(-\dfrac{19}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{72}{7}\right)\sqrt{4}+\left(\dfrac{19}{8}\right)\sqrt{18}+\left(\dfrac{7}{4}\right)\sqrt{4}\right)-\left(\left(\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{50}{3}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{80}{3}\right)\sqrt{8}\right)-\dfrac{23}{5}\right)\right)\\ &=&\left(\left(-\dfrac{17}{3}\right)\sqrt{2}+\dfrac{15}{2}-8-\left(\left(-\dfrac{17}{3}\right)\sqrt{2}\right)-\left(\left(4\right)\sqrt{2}\right)-\left(\left(27\right)\sqrt{2}\right)-9-\left(\left(47\right)\sqrt{2}\right)+\dfrac{48}{7}+\dfrac{3}{7}\right)\times\left(\left(\left(\dfrac{5}{2}\right)\sqrt{2}+\left(-\dfrac{135}{2}\right)\sqrt{2}+\left(\dfrac{74}{5}\right)\sqrt{2}-\dfrac{53}{4}+\left(-\dfrac{57}{2}\right)\sqrt{2}\right)-\left(\dfrac{144}{7}+\left(\dfrac{57}{8}\right)\sqrt{2}+\dfrac{7}{2}\right)-\left(33+\dfrac{100}{3}\right)-\left(\left(\left(\dfrac{160}{3}\right)\sqrt{2}\right)-\dfrac{23}{5}\right)\right)\\ &=&\left(\left(-78\right)\sqrt{2}-\dfrac{31}{14}\right)\left(\left(-\dfrac{16699}{120}\right)\sqrt{2}-\dfrac{41603}{420}\right)\\ &=&\left(\dfrac{217087}{20}\right)\sqrt{4}+\left(\dfrac{2699561}{336}\right)\sqrt{2}+\dfrac{1289693}{5880}\\ &=&\dfrac{128936849}{5880}+\left(\dfrac{2699561}{336}\right)\sqrt{2}\\ \end{eqnarray*}