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(-\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{67}{3}\right)\sqrt{125}+\left(7\right)\sqrt{25}+\left(-\dfrac{7}{3}\right)\sqrt{125}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{28}{5}\right)\sqrt{45}-\dfrac{25}{7}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{45}+\left(-\dfrac{27}{4}\right)\sqrt{20}+\dfrac{25}{4}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-8\right)\sqrt{125}+\left(\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\dfrac{67}{4}-\left(\left(\dfrac{32}{9}\right)\sqrt{25}\right)\right)$ et $ Y=\left(\left(2\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{25}\right)+2\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{77}{6}\right)\sqrt{25}+\left(\dfrac{67}{3}\right)\sqrt{125}+\left(7\right)\sqrt{25}+\left(-\dfrac{7}{3}\right)\sqrt{125}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{28}{5}\right)\sqrt{45}-\dfrac{25}{7}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{45}+\left(-\dfrac{27}{4}\right)\sqrt{20}+\dfrac{25}{4}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-8\right)\sqrt{125}+\left(\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\dfrac{67}{4}-\left(\left(\dfrac{32}{9}\right)\sqrt{25}\right)\right)\right)+\left(\left(\left(2\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{25}\right)+2\right)\right)\\ &=&\left(\left(-\dfrac{385}{6}+\left(\dfrac{335}{3}\right)\sqrt{5}+35+\left(-\dfrac{35}{3}\right)\sqrt{5}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{84}{5}\right)\sqrt{5}-\dfrac{25}{7}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-17\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\dfrac{25}{4}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-40\right)\sqrt{5}+\left(\dfrac{3}{4}\right)\sqrt{5}\right)-\left(\dfrac{67}{4}-\dfrac{160}{9}\right)\right)+\left(\left(\left(6\right)\sqrt{5}\right)-\left(\left(\left(39\right)\sqrt{5}\right)+\dfrac{315}{8}+2\right)\right)\\ &=&\left(-\dfrac{385}{6}+\left(\dfrac{335}{3}\right)\sqrt{5}+35+\left(-\dfrac{35}{3}\right)\sqrt{5}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{84}{5}\right)\sqrt{5}-\dfrac{25}{7}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-17\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\dfrac{25}{4}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-40\right)\sqrt{5}+\left(\dfrac{3}{4}\right)\sqrt{5}\right)-\left(\dfrac{67}{4}-\dfrac{160}{9}\right)+\left(\left(6\right)\sqrt{5}\right)-\left(\left(\left(39\right)\sqrt{5}\right)+\dfrac{315}{8}+2\right)\\ &=&-\dfrac{41299}{504}+\left(\dfrac{6817}{60}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{67}{3}\right)\sqrt{125}+\left(7\right)\sqrt{25}+\left(-\dfrac{7}{3}\right)\sqrt{125}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{28}{5}\right)\sqrt{45}-\dfrac{25}{7}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{45}+\left(-\dfrac{27}{4}\right)\sqrt{20}+\dfrac{25}{4}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-8\right)\sqrt{125}+\left(\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\dfrac{67}{4}-\left(\left(\dfrac{32}{9}\right)\sqrt{25}\right)\right)\right)-\left(\left(\left(2\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{25}\right)+2\right)\right)\\ &=&\left(\left(-\dfrac{385}{6}+\left(\dfrac{335}{3}\right)\sqrt{5}+35+\left(-\dfrac{35}{3}\right)\sqrt{5}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{84}{5}\right)\sqrt{5}-\dfrac{25}{7}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-17\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\dfrac{25}{4}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-40\right)\sqrt{5}+\left(\dfrac{3}{4}\right)\sqrt{5}\right)-\left(\dfrac{67}{4}-\dfrac{160}{9}\right)\right)-\left(\left(\left(6\right)\sqrt{5}\right)-\left(\left(\left(39\right)\sqrt{5}\right)+\dfrac{315}{8}+2\right)\right)\\ &=&\left(-\dfrac{10223}{252}+\left(\dfrac{8797}{60}\right)\sqrt{5}\right)-\left(\left(-33\right)\sqrt{5}-\dfrac{331}{8}\right)\\ &=&-\dfrac{10223}{252}+\left(\dfrac{8797}{60}\right)\sqrt{5}+\left(33\right)\sqrt{5}+\dfrac{331}{8}\\ &=&\dfrac{407}{504}+\left(\dfrac{10777}{60}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{77}{6}\right)\sqrt{25}+\left(\dfrac{67}{3}\right)\sqrt{125}+\left(7\right)\sqrt{25}+\left(-\dfrac{7}{3}\right)\sqrt{125}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{28}{5}\right)\sqrt{45}-\dfrac{25}{7}+\left(\dfrac{35}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{17}{3}\right)\sqrt{45}+\left(-\dfrac{27}{4}\right)\sqrt{20}+\dfrac{25}{4}+\left(-\dfrac{7}{3}\right)\sqrt{125}+\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-8\right)\sqrt{125}+\left(\dfrac{1}{4}\right)\sqrt{45}\right)-\left(\dfrac{67}{4}-\left(\left(\dfrac{32}{9}\right)\sqrt{25}\right)\right)\right)\times\left(\left(\left(2\right)\sqrt{45}\right)-\left(\left(\left(\dfrac{39}{2}\right)\sqrt{20}\right)-\left(\left(-\dfrac{63}{8}\right)\sqrt{25}\right)+2\right)\right)\\ &=&\left(\left(-\dfrac{385}{6}+\left(\dfrac{335}{3}\right)\sqrt{5}+35+\left(-\dfrac{35}{3}\right)\sqrt{5}-2\right)-\left(\dfrac{39}{4}-2+\left(\dfrac{84}{5}\right)\sqrt{5}-\dfrac{25}{7}+\left(\dfrac{70}{3}\right)\sqrt{5}\right)-\left(\left(-17\right)\sqrt{5}+\left(-\dfrac{27}{2}\right)\sqrt{5}+\dfrac{25}{4}+\left(-\dfrac{35}{3}\right)\sqrt{5}+\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-40\right)\sqrt{5}+\left(\dfrac{3}{4}\right)\sqrt{5}\right)-\left(\dfrac{67}{4}-\dfrac{160}{9}\right)\right)\times\left(\left(\left(6\right)\sqrt{5}\right)-\left(\left(\left(39\right)\sqrt{5}\right)+\dfrac{315}{8}+2\right)\right)\\ &=&\left(-\dfrac{10223}{252}+\left(\dfrac{8797}{60}\right)\sqrt{5}\right)\left(\left(-33\right)\sqrt{5}-\dfrac{331}{8}\right)\\ &=&\left(-\dfrac{5294843}{1120}\right)\sqrt{5}+\dfrac{3383813}{2016}+\left(-\dfrac{96767}{20}\right)\sqrt{25}\\ &=&\left(-\dfrac{5294843}{1120}\right)\sqrt{5}-\dfrac{45386755}{2016}\\ \end{eqnarray*}