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{7}{3}+\left(8\right)\sqrt{50}+\left(-\dfrac{9}{2}\right)\sqrt{18}+\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{18}\right)\right)$ et $ Y=\left(\dfrac{64}{5}\right)\sqrt{50}$ . 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{7}{3}+\left(8\right)\sqrt{50}+\left(-\dfrac{9}{2}\right)\sqrt{18}+\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{18}\right)\right)\right)+\left(\left(\dfrac{64}{5}\right)\sqrt{50}\right)\\ &=&\left(\left(-\dfrac{7}{3}+\left(40\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)-\left(\left(77\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)\right)-\dfrac{66}{5}-\left(\left(\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{2}\right)\right)\right)+\left(\left(64\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{7}{3}+\left(40\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)-\left(\left(77\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)\right)-\dfrac{66}{5}-\left(\left(\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{2}\right)\right)+\left(64\right)\sqrt{2}\\ &=&-\dfrac{233}{15}+\left(\dfrac{5783}{84}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(-\dfrac{7}{3}+\left(8\right)\sqrt{50}+\left(-\dfrac{9}{2}\right)\sqrt{18}+\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{18}\right)\right)\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{50}\right)\\ &=&\left(\left(-\dfrac{7}{3}+\left(40\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)-\left(\left(77\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)\right)-\dfrac{66}{5}-\left(\left(\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{2}\right)\right)\right)-\left(\left(64\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{233}{15}+\left(\dfrac{407}{84}\right)\sqrt{2}\right)-\left(\left(64\right)\sqrt{2}\right)\\ &=&-\dfrac{233}{15}+\left(\dfrac{407}{84}\right)\sqrt{2}+\left(-64\right)\sqrt{2}\\ &=&-\dfrac{233}{15}+\left(-\dfrac{4969}{84}\right)\sqrt{2}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(-\dfrac{7}{3}+\left(8\right)\sqrt{50}+\left(-\dfrac{9}{2}\right)\sqrt{18}+\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{18}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{18}\right)-\left(\left(\dfrac{50}{3}\right)\sqrt{8}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{50}\right)-\left(\left(-\dfrac{63}{5}\right)\sqrt{50}\right)\right)-\left(\left(\dfrac{33}{5}\right)\sqrt{4}\right)-\left(\left(\left(\dfrac{5}{2}\right)\sqrt{8}\right)-\left(\left(-\dfrac{7}{8}\right)\sqrt{8}\right)-\left(\left(-\dfrac{81}{7}\right)\sqrt{8}\right)-\left(\left(\dfrac{13}{7}\right)\sqrt{18}\right)\right)\right)\times\left(\left(\dfrac{64}{5}\right)\sqrt{50}\right)\\ &=&\left(\left(-\dfrac{7}{3}+\left(40\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}+\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{55}{3}\right)\sqrt{2}\right)-\left(\left(\left(-\dfrac{27}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{100}{3}\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)-\left(\left(77\right)\sqrt{2}\right)-\left(\left(-63\right)\sqrt{2}\right)\right)-\dfrac{66}{5}-\left(\left(\left(5\right)\sqrt{2}\right)-\left(\left(-\dfrac{7}{4}\right)\sqrt{2}\right)-\left(\left(-\dfrac{162}{7}\right)\sqrt{2}\right)-\left(\left(\dfrac{39}{7}\right)\sqrt{2}\right)\right)\right)\times\left(\left(64\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{233}{15}+\left(\dfrac{407}{84}\right)\sqrt{2}\right)\left(\left(64\right)\sqrt{2}\right)\\ &=&\left(-\dfrac{14912}{15}\right)\sqrt{2}+\left(\dfrac{6512}{21}\right)\sqrt{4}\\ &=&\left(-\dfrac{14912}{15}\right)\sqrt{2}+\dfrac{13024}{21}\\ \end{eqnarray*}