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(-3\right)\sqrt{125}+\left(\dfrac{14}{3}\right)\sqrt{45}+\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(8\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{5}{7}\right)\sqrt{45}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{25}\right)\right)$ et $ Y=\left(\dfrac{10}{7}+\left(-\dfrac{65}{6}\right)\sqrt{45}+\left(\dfrac{66}{5}\right)\sqrt{25}+\left(\dfrac{75}{8}\right)\sqrt{125}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{25}{7}\right)\sqrt{20}\right)\right)-\left(\left(5\right)\sqrt{125}+\dfrac{73}{2}+\left(-\dfrac{49}{8}\right)\sqrt{25}\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(-3\right)\sqrt{125}+\left(\dfrac{14}{3}\right)\sqrt{45}+\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(8\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{5}{7}\right)\sqrt{45}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{25}\right)\right)\right)+\left(\left(\dfrac{10}{7}+\left(-\dfrac{65}{6}\right)\sqrt{45}+\left(\dfrac{66}{5}\right)\sqrt{25}+\left(\dfrac{75}{8}\right)\sqrt{125}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{25}{7}\right)\sqrt{20}\right)\right)-\left(\left(5\right)\sqrt{125}+\dfrac{73}{2}+\left(-\dfrac{49}{8}\right)\sqrt{25}\right)\right)\\ &=&\left(\left(\left(-15\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-5\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{15}{7}\right)\sqrt{5}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-5\right)\sqrt{5}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{5}\right)+42+\dfrac{35}{6}\right)\right)+\left(\left(\dfrac{10}{7}+\left(-\dfrac{65}{2}\right)\sqrt{5}+66+\left(\dfrac{375}{8}\right)\sqrt{5}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{50}{7}\right)\sqrt{5}\right)\right)-\left(\left(25\right)\sqrt{5}+\dfrac{73}{2}-\dfrac{245}{8}\right)\right)\\ &=&\left(\left(-15\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-5\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{15}{7}\right)\sqrt{5}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-5\right)\sqrt{5}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{5}\right)+42+\dfrac{35}{6}\right)+\left(\dfrac{10}{7}+\left(-\dfrac{65}{2}\right)\sqrt{5}+66+\left(\dfrac{375}{8}\right)\sqrt{5}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{50}{7}\right)\sqrt{5}\right)\right)-\left(\left(25\right)\sqrt{5}+\dfrac{73}{2}-\dfrac{245}{8}\right)\\ &=&\left(-\dfrac{6313}{168}\right)\sqrt{5}+\dfrac{5263}{504}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-3\right)\sqrt{125}+\left(\dfrac{14}{3}\right)\sqrt{45}+\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(8\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{5}{7}\right)\sqrt{45}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{25}\right)\right)\right)-\left(\left(\dfrac{10}{7}+\left(-\dfrac{65}{6}\right)\sqrt{45}+\left(\dfrac{66}{5}\right)\sqrt{25}+\left(\dfrac{75}{8}\right)\sqrt{125}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{25}{7}\right)\sqrt{20}\right)\right)-\left(\left(5\right)\sqrt{125}+\dfrac{73}{2}+\left(-\dfrac{49}{8}\right)\sqrt{25}\right)\right)\\ &=&\left(\left(\left(-15\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-5\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{15}{7}\right)\sqrt{5}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-5\right)\sqrt{5}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{5}\right)+42+\dfrac{35}{6}\right)\right)-\left(\left(\dfrac{10}{7}+\left(-\dfrac{65}{2}\right)\sqrt{5}+66+\left(\dfrac{375}{8}\right)\sqrt{5}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{50}{7}\right)\sqrt{5}\right)\right)-\left(\left(25\right)\sqrt{5}+\dfrac{73}{2}-\dfrac{245}{8}\right)\right)\\ &=&\left(\left(-\dfrac{416}{21}\right)\sqrt{5}-\dfrac{893}{18}\right)-\left(\dfrac{3363}{56}+\left(-\dfrac{995}{56}\right)\sqrt{5}\right)\\ &=&\left(-\dfrac{416}{21}\right)\sqrt{5}-\dfrac{893}{18}+-\dfrac{3363}{56}+\left(\dfrac{995}{56}\right)\sqrt{5}\\ &=&\left(-\dfrac{49}{24}\right)\sqrt{5}-\dfrac{55271}{504}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-3\right)\sqrt{125}+\left(\dfrac{14}{3}\right)\sqrt{45}+\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(8\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{5}{7}\right)\sqrt{45}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-\dfrac{5}{3}\right)\sqrt{45}\right)-\left(\left(\dfrac{11}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{42}{5}\right)\sqrt{25}\right)-\left(\left(-\dfrac{7}{6}\right)\sqrt{25}\right)\right)\right)\times\left(\left(\dfrac{10}{7}+\left(-\dfrac{65}{6}\right)\sqrt{45}+\left(\dfrac{66}{5}\right)\sqrt{25}+\left(\dfrac{75}{8}\right)\sqrt{125}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{25}{7}\right)\sqrt{20}\right)\right)-\left(\left(5\right)\sqrt{125}+\dfrac{73}{2}+\left(-\dfrac{49}{8}\right)\sqrt{25}\right)\right)\\ &=&\left(\left(\left(-15\right)\sqrt{5}+\left(14\right)\sqrt{5}+\left(-5\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)-\left(\left(\left(\dfrac{15}{7}\right)\sqrt{5}\right)+\dfrac{8}{9}+\dfrac{8}{9}\right)-\left(\left(\left(-5\right)\sqrt{5}\right)-\left(\left(\dfrac{22}{3}\right)\sqrt{5}\right)+42+\dfrac{35}{6}\right)\right)\times\left(\left(\dfrac{10}{7}+\left(-\dfrac{65}{2}\right)\sqrt{5}+66+\left(\dfrac{375}{8}\right)\sqrt{5}\right)-\left(\dfrac{3}{2}-\left(\left(-\dfrac{50}{7}\right)\sqrt{5}\right)\right)-\left(\left(25\right)\sqrt{5}+\dfrac{73}{2}-\dfrac{245}{8}\right)\right)\\ &=&\left(\left(-\dfrac{416}{21}\right)\sqrt{5}-\dfrac{893}{18}\right)\left(\dfrac{3363}{56}+\left(-\dfrac{995}{56}\right)\sqrt{5}\right)\\ &=&\left(-\dfrac{2174303}{7056}\right)\sqrt{5}+\left(\dfrac{51740}{147}\right)\sqrt{25}-\dfrac{1001053}{336}\\ &=&\left(-\dfrac{2174303}{7056}\right)\sqrt{5}-\dfrac{956057}{784}\\ \end{eqnarray*}