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{17}{9}\right)\sqrt{125}\right)-\dfrac{47}{2}-\left(\left(-\dfrac{35}{3}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{125}+\left(3\right)\sqrt{25}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(3\right)\sqrt{45}\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{9}\right)\sqrt{125}\right)-3\right)$ et $ Y=\left(\left(\dfrac{33}{2}\right)\sqrt{45}-5-\dfrac{15}{2}+\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{13}{8}\right)\sqrt{125}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{20}\right)+9\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}+\left(1\right)\sqrt{125}+\left(-\dfrac{53}{2}\right)\sqrt{125}+\left(\dfrac{2}{3}\right)\sqrt{125}+\left(-\dfrac{1}{9}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)\right)-\dfrac{8}{3}$ . 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{17}{9}\right)\sqrt{125}\right)-\dfrac{47}{2}-\left(\left(-\dfrac{35}{3}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{125}+\left(3\right)\sqrt{25}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(3\right)\sqrt{45}\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{9}\right)\sqrt{125}\right)-3\right)\right)+\left(\left(\left(\dfrac{33}{2}\right)\sqrt{45}-5-\dfrac{15}{2}+\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{13}{8}\right)\sqrt{125}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{20}\right)+9\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}+\left(1\right)\sqrt{125}+\left(-\dfrac{53}{2}\right)\sqrt{125}+\left(\dfrac{2}{3}\right)\sqrt{125}+\left(-\dfrac{1}{9}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)\right)-\dfrac{8}{3}\right)\\ &=&\left(\left(\left(\left(-\dfrac{85}{9}\right)\sqrt{5}\right)-\dfrac{47}{2}-\left(\left(-35\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{400}{9}\right)\sqrt{5}+15\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(9\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{5}\right)-3\right)\right)+\left(\left(\left(\dfrac{99}{2}\right)\sqrt{5}-5-\dfrac{15}{2}+\left(-\dfrac{265}{2}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{65}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{154}{5}\right)\sqrt{5}\right)+9\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}+\left(5\right)\sqrt{5}+\left(-\dfrac{265}{2}\right)\sqrt{5}+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{5}{9}\right)-\left(\left(\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{2}\right)\sqrt{5}\right)+30+\dfrac{225}{8}-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)\right)-\dfrac{8}{3}\right)\\ &=&\left(\left(\left(-\dfrac{85}{9}\right)\sqrt{5}\right)-\dfrac{47}{2}-\left(\left(-35\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{400}{9}\right)\sqrt{5}+15\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(9\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{5}\right)-3\right)+\left(\left(\dfrac{99}{2}\right)\sqrt{5}-5-\dfrac{15}{2}+\left(-\dfrac{265}{2}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{65}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{154}{5}\right)\sqrt{5}\right)+9\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}+\left(5\right)\sqrt{5}+\left(-\dfrac{265}{2}\right)\sqrt{5}+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{5}{9}\right)-\left(\left(\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{2}\right)\sqrt{5}\right)+30+\dfrac{225}{8}-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)\right)-\dfrac{8}{3}\\ &=&\left(\dfrac{32443}{1260}\right)\sqrt{5}-\dfrac{8441}{72}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\left(-\dfrac{17}{9}\right)\sqrt{125}\right)-\dfrac{47}{2}-\left(\left(-\dfrac{35}{3}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{125}+\left(3\right)\sqrt{25}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(3\right)\sqrt{45}\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{9}\right)\sqrt{125}\right)-3\right)\right)-\left(\left(\left(\dfrac{33}{2}\right)\sqrt{45}-5-\dfrac{15}{2}+\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{13}{8}\right)\sqrt{125}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{20}\right)+9\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}+\left(1\right)\sqrt{125}+\left(-\dfrac{53}{2}\right)\sqrt{125}+\left(\dfrac{2}{3}\right)\sqrt{125}+\left(-\dfrac{1}{9}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)\right)-\dfrac{8}{3}\right)\\ &=&\left(\left(\left(\left(-\dfrac{85}{9}\right)\sqrt{5}\right)-\dfrac{47}{2}-\left(\left(-35\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{400}{9}\right)\sqrt{5}+15\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(9\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{5}\right)-3\right)\right)-\left(\left(\left(\dfrac{99}{2}\right)\sqrt{5}-5-\dfrac{15}{2}+\left(-\dfrac{265}{2}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{65}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{154}{5}\right)\sqrt{5}\right)+9\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}+\left(5\right)\sqrt{5}+\left(-\dfrac{265}{2}\right)\sqrt{5}+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{5}{9}\right)-\left(\left(\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{2}\right)\sqrt{5}\right)+30+\dfrac{225}{8}-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)\right)-\dfrac{8}{3}\right)\\ &=&\left(\left(\dfrac{6697}{72}\right)\sqrt{5}-\dfrac{71}{2}\right)-\left(\left(-\dfrac{56503}{840}\right)\sqrt{5}-\dfrac{5885}{72}\right)\\ &=&\left(\dfrac{6697}{72}\right)\sqrt{5}-\dfrac{71}{2}+\left(\dfrac{56503}{840}\right)\sqrt{5}+\dfrac{5885}{72}\\ &=&\left(\dfrac{50488}{315}\right)\sqrt{5}+\dfrac{3329}{72}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\left(-\dfrac{17}{9}\right)\sqrt{125}\right)-\dfrac{47}{2}-\left(\left(-\dfrac{35}{3}\right)\sqrt{45}\right)\right)-\left(\left(-\dfrac{80}{9}\right)\sqrt{125}+\left(3\right)\sqrt{25}\right)-\left(\left(\left(-9\right)\sqrt{45}\right)-\left(\left(3\right)\sqrt{45}\right)-\left(\left(-\dfrac{11}{8}\right)\sqrt{125}\right)-\left(\left(-\dfrac{11}{9}\right)\sqrt{125}\right)-3\right)\right)\times\left(\left(\left(\dfrac{33}{2}\right)\sqrt{45}-5-\dfrac{15}{2}+\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(\left(-\dfrac{13}{8}\right)\sqrt{125}\right)-\left(\left(\dfrac{77}{5}\right)\sqrt{20}\right)+9\right)-\left(\left(-\dfrac{53}{6}\right)\sqrt{20}+\left(1\right)\sqrt{125}+\left(-\dfrac{53}{2}\right)\sqrt{125}+\left(\dfrac{2}{3}\right)\sqrt{125}+\left(-\dfrac{1}{9}\right)\sqrt{25}\right)-\left(\left(\left(-\dfrac{8}{3}\right)\sqrt{20}\right)-\left(\left(-\dfrac{53}{2}\right)\sqrt{125}\right)-\left(\left(-6\right)\sqrt{25}\right)-\left(\left(-\dfrac{45}{8}\right)\sqrt{25}\right)-\left(\left(-\dfrac{53}{7}\right)\sqrt{125}\right)\right)-\dfrac{8}{3}\right)\\ &=&\left(\left(\left(\left(-\dfrac{85}{9}\right)\sqrt{5}\right)-\dfrac{47}{2}-\left(\left(-35\right)\sqrt{5}\right)\right)-\left(\left(-\dfrac{400}{9}\right)\sqrt{5}+15\right)-\left(\left(\left(-27\right)\sqrt{5}\right)-\left(\left(9\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{8}\right)\sqrt{5}\right)-\left(\left(-\dfrac{55}{9}\right)\sqrt{5}\right)-3\right)\right)\times\left(\left(\left(\dfrac{99}{2}\right)\sqrt{5}-5-\dfrac{15}{2}+\left(-\dfrac{265}{2}\right)\sqrt{5}\right)-\left(\left(\left(-\dfrac{65}{8}\right)\sqrt{5}\right)-\left(\left(\dfrac{154}{5}\right)\sqrt{5}\right)+9\right)-\left(\left(-\dfrac{53}{3}\right)\sqrt{5}+\left(5\right)\sqrt{5}+\left(-\dfrac{265}{2}\right)\sqrt{5}+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{5}{9}\right)-\left(\left(\left(-\dfrac{16}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{265}{2}\right)\sqrt{5}\right)+30+\dfrac{225}{8}-\left(\left(-\dfrac{265}{7}\right)\sqrt{5}\right)\right)-\dfrac{8}{3}\right)\\ &=&\left(\left(\dfrac{6697}{72}\right)\sqrt{5}-\dfrac{71}{2}\right)\left(\left(-\dfrac{56503}{840}\right)\sqrt{5}-\dfrac{5885}{72}\right)\\ &=&\left(-\dfrac{378400591}{60480}\right)\sqrt{25}+\left(-\dfrac{45415179408}{8709120}\right)\sqrt{5}+\dfrac{417835}{144}\\ &=&-\dfrac{49435552944}{1741824}+\left(-\dfrac{45415179408}{8709120}\right)\sqrt{5}\\ \end{eqnarray*}