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(\left(\left(-\dfrac{25}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{4}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{4}\right)\right)-\left(\left(\left(8\right)\sqrt{4}\right)+8-\left(\left(-\dfrac{7}{5}\right)\sqrt{18}\right)\right)-\left(\left(4\right)\sqrt{8}-\dfrac{22}{3}+\left(-\dfrac{58}{9}\right)\sqrt{8}+\left(-\dfrac{17}{9}\right)\sqrt{8}+\left(1\right)\sqrt{8}\right)-\left(\left(-4\right)\sqrt{8}\right)$ et $ Y=\left(\left(0\right)\sqrt{8}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{25}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{18}\right)+\left(-\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{62}{3}\right)\sqrt{50}+\left(9\right)\sqrt{18}-5+\left(\left(-8\right)\sqrt{8}\right)-5-\left(\left(\dfrac{11}{4}\right)\sqrt{8}\right)+\left(\left(\dfrac{43}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{50}\right)-\dfrac{15}{4}-\left(\left(9\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}\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{25}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{4}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{4}\right)\right)-\left(\left(\left(8\right)\sqrt{4}\right)+8-\left(\left(-\dfrac{7}{5}\right)\sqrt{18}\right)\right)-\left(\left(4\right)\sqrt{8}-\dfrac{22}{3}+\left(-\dfrac{58}{9}\right)\sqrt{8}+\left(-\dfrac{17}{9}\right)\sqrt{8}+\left(1\right)\sqrt{8}\right)-\left(\left(-4\right)\sqrt{8}\right)\right)+\left(\left(\left(0\right)\sqrt{8}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{25}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{18}\right)+\left(-\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{62}{3}\right)\sqrt{50}+\left(9\right)\sqrt{18}-5+\left(\left(-8\right)\sqrt{8}\right)-5-\left(\left(\dfrac{11}{4}\right)\sqrt{8}\right)+\left(\left(\dfrac{43}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{50}\right)-\dfrac{15}{4}-\left(\left(9\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}\right)\right)\\ &=&\left(\left(-\dfrac{25}{4}-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\dfrac{4}{9}-10-\dfrac{7}{3}\right)-\left(16+8-\left(\left(-\dfrac{21}{5}\right)\sqrt{2}\right)\right)-\left(\left(8\right)\sqrt{2}-\dfrac{22}{3}+\left(-\dfrac{116}{9}\right)\sqrt{2}+\left(-\dfrac{34}{9}\right)\sqrt{2}+\left(2\right)\sqrt{2}\right)-\left(\left(-8\right)\sqrt{2}\right)\right)+\left(\left(\left(0\right)\sqrt{2}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{125}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{2}\right)+\left(-\dfrac{80}{7}\right)\sqrt{2}+\left(-\dfrac{310}{3}\right)\sqrt{2}+\left(27\right)\sqrt{2}-5+\left(\left(-16\right)\sqrt{2}\right)-5-\left(\left(\dfrac{11}{2}\right)\sqrt{2}\right)+\left(\left(43\right)\sqrt{2}\right)-\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)-\dfrac{15}{4}-\left(\left(27\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}\right)\right)\\ &=&\left(-\dfrac{25}{4}-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\dfrac{4}{9}-10-\dfrac{7}{3}\right)-\left(16+8-\left(\left(-\dfrac{21}{5}\right)\sqrt{2}\right)\right)-\left(\left(8\right)\sqrt{2}-\dfrac{22}{3}+\left(-\dfrac{116}{9}\right)\sqrt{2}+\left(-\dfrac{34}{9}\right)\sqrt{2}+\left(2\right)\sqrt{2}\right)-\left(\left(-8\right)\sqrt{2}\right)+\left(\left(0\right)\sqrt{2}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{125}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{2}\right)+\left(-\dfrac{80}{7}\right)\sqrt{2}+\left(-\dfrac{310}{3}\right)\sqrt{2}+\left(27\right)\sqrt{2}-5+\left(\left(-16\right)\sqrt{2}\right)-5-\left(\left(\dfrac{11}{2}\right)\sqrt{2}\right)+\left(\left(43\right)\sqrt{2}\right)-\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)-\dfrac{15}{4}-\left(\left(27\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}\right)\\ &=&-\dfrac{571}{9}+\left(-\dfrac{49339}{420}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\left(-\dfrac{25}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{4}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{4}\right)\right)-\left(\left(\left(8\right)\sqrt{4}\right)+8-\left(\left(-\dfrac{7}{5}\right)\sqrt{18}\right)\right)-\left(\left(4\right)\sqrt{8}-\dfrac{22}{3}+\left(-\dfrac{58}{9}\right)\sqrt{8}+\left(-\dfrac{17}{9}\right)\sqrt{8}+\left(1\right)\sqrt{8}\right)-\left(\left(-4\right)\sqrt{8}\right)\right)-\left(\left(\left(0\right)\sqrt{8}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{25}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{18}\right)+\left(-\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{62}{3}\right)\sqrt{50}+\left(9\right)\sqrt{18}-5+\left(\left(-8\right)\sqrt{8}\right)-5-\left(\left(\dfrac{11}{4}\right)\sqrt{8}\right)+\left(\left(\dfrac{43}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{50}\right)-\dfrac{15}{4}-\left(\left(9\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}\right)\right)\\ &=&\left(\left(-\dfrac{25}{4}-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\dfrac{4}{9}-10-\dfrac{7}{3}\right)-\left(16+8-\left(\left(-\dfrac{21}{5}\right)\sqrt{2}\right)\right)-\left(\left(8\right)\sqrt{2}-\dfrac{22}{3}+\left(-\dfrac{116}{9}\right)\sqrt{2}+\left(-\dfrac{34}{9}\right)\sqrt{2}+\left(2\right)\sqrt{2}\right)-\left(\left(-8\right)\sqrt{2}\right)\right)-\left(\left(\left(0\right)\sqrt{2}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{125}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{2}\right)+\left(-\dfrac{80}{7}\right)\sqrt{2}+\left(-\dfrac{310}{3}\right)\sqrt{2}+\left(27\right)\sqrt{2}-5+\left(\left(-16\right)\sqrt{2}\right)-5-\left(\left(\dfrac{11}{2}\right)\sqrt{2}\right)+\left(\left(43\right)\sqrt{2}\right)-\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)-\dfrac{15}{4}-\left(\left(27\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}\right)\right)\\ &=&\left(-\dfrac{1253}{36}+\left(-\dfrac{1711}{30}\right)\sqrt{2}\right)-\left(\left(-\dfrac{5077}{84}\right)\sqrt{2}-\dfrac{1031}{36}\right)\\ &=&-\dfrac{1253}{36}+\left(-\dfrac{1711}{30}\right)\sqrt{2}+\left(\dfrac{5077}{84}\right)\sqrt{2}+\dfrac{1031}{36}\\ &=&-\dfrac{37}{6}+\left(\dfrac{477}{140}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\left(-\dfrac{25}{8}\right)\sqrt{4}\right)-\left(\left(\dfrac{27}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{2}{9}\right)\sqrt{4}\right)-\left(\left(5\right)\sqrt{4}\right)-\left(\left(\dfrac{7}{6}\right)\sqrt{4}\right)\right)-\left(\left(\left(8\right)\sqrt{4}\right)+8-\left(\left(-\dfrac{7}{5}\right)\sqrt{18}\right)\right)-\left(\left(4\right)\sqrt{8}-\dfrac{22}{3}+\left(-\dfrac{58}{9}\right)\sqrt{8}+\left(-\dfrac{17}{9}\right)\sqrt{8}+\left(1\right)\sqrt{8}\right)-\left(\left(-4\right)\sqrt{8}\right)\right)\times\left(\left(\left(0\right)\sqrt{8}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{25}{2}\right)\sqrt{50}\right)-\left(\left(-\dfrac{13}{7}\right)\sqrt{18}\right)+\left(-\dfrac{40}{7}\right)\sqrt{8}+\left(-\dfrac{62}{3}\right)\sqrt{50}+\left(9\right)\sqrt{18}-5+\left(\left(-8\right)\sqrt{8}\right)-5-\left(\left(\dfrac{11}{4}\right)\sqrt{8}\right)+\left(\left(\dfrac{43}{3}\right)\sqrt{18}\right)-\left(\left(\dfrac{21}{4}\right)\sqrt{50}\right)-\dfrac{15}{4}-\left(\left(9\right)\sqrt{18}\right)-\left(\left(3\right)\sqrt{18}\right)\right)\\ &=&\left(\left(-\dfrac{25}{4}-\left(\left(\dfrac{135}{2}\right)\sqrt{2}\right)+\dfrac{4}{9}-10-\dfrac{7}{3}\right)-\left(16+8-\left(\left(-\dfrac{21}{5}\right)\sqrt{2}\right)\right)-\left(\left(8\right)\sqrt{2}-\dfrac{22}{3}+\left(-\dfrac{116}{9}\right)\sqrt{2}+\left(-\dfrac{34}{9}\right)\sqrt{2}+\left(2\right)\sqrt{2}\right)-\left(\left(-8\right)\sqrt{2}\right)\right)\times\left(\left(\left(0\right)\sqrt{2}\right)-7-\dfrac{71}{9}-\left(\left(-\dfrac{125}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{39}{7}\right)\sqrt{2}\right)+\left(-\dfrac{80}{7}\right)\sqrt{2}+\left(-\dfrac{310}{3}\right)\sqrt{2}+\left(27\right)\sqrt{2}-5+\left(\left(-16\right)\sqrt{2}\right)-5-\left(\left(\dfrac{11}{2}\right)\sqrt{2}\right)+\left(\left(43\right)\sqrt{2}\right)-\left(\left(\dfrac{105}{4}\right)\sqrt{2}\right)-\dfrac{15}{4}-\left(\left(27\right)\sqrt{2}\right)-\left(\left(9\right)\sqrt{2}\right)\right)\\ &=&\left(-\dfrac{1253}{36}+\left(-\dfrac{1711}{30}\right)\sqrt{2}\right)\left(\left(-\dfrac{5077}{84}\right)\sqrt{2}-\dfrac{1031}{36}\right)\\ &=&\left(\dfrac{8071997}{2160}\right)\sqrt{2}+\dfrac{1291843}{1296}+\left(\dfrac{8686747}{2520}\right)\sqrt{4}\\ &=&\left(\dfrac{8071997}{2160}\right)\sqrt{2}+\dfrac{12885746292}{1632960}\\ \end{eqnarray*}