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{75}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{20}\right)+\left(-\dfrac{17}{2}\right)\sqrt{125}+\left(\dfrac{76}{3}\right)\sqrt{20}+\left(8\right)\sqrt{45}+\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{125}+\left(-2\right)\sqrt{20}-\dfrac{34}{9}$ et $ Y=\left(\left(-\dfrac{23}{7}\right)\sqrt{20}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{20}\right)-\dfrac{76}{7}-\left(\left(\dfrac{21}{5}\right)\sqrt{20}\right)+\left(-\dfrac{15}{8}\right)\sqrt{125}+\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{13}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{4}{7}\right)\sqrt{45}\right)-\dfrac{3}{5}$ . 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{75}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{20}\right)+\left(-\dfrac{17}{2}\right)\sqrt{125}+\left(\dfrac{76}{3}\right)\sqrt{20}+\left(8\right)\sqrt{45}+\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{125}+\left(-2\right)\sqrt{20}-\dfrac{34}{9}\right)+\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{20}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{20}\right)-\dfrac{76}{7}-\left(\left(\dfrac{21}{5}\right)\sqrt{20}\right)+\left(-\dfrac{15}{8}\right)\sqrt{125}+\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{13}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{4}{7}\right)\sqrt{45}\right)-\dfrac{3}{5}\right)\\ &=&\left(\dfrac{375}{4}-\left(\left(-4\right)\sqrt{5}\right)+\left(-\dfrac{85}{2}\right)\sqrt{5}+\left(\dfrac{152}{3}\right)\sqrt{5}+\left(24\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-\dfrac{215}{7}\right)\sqrt{5}+\left(-4\right)\sqrt{5}-\dfrac{34}{9}\right)+\left(\left(\left(-\dfrac{46}{7}\right)\sqrt{5}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{128}{5}\right)\sqrt{5}\right)-\dfrac{76}{7}-\left(\left(\dfrac{42}{5}\right)\sqrt{5}\right)+\left(-\dfrac{75}{8}\right)\sqrt{5}+\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(13\right)\sqrt{5}\right)+\left(\left(-18\right)\sqrt{5}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{5}\right)-\dfrac{3}{5}\right)\\ &=&\dfrac{375}{4}-\left(\left(-4\right)\sqrt{5}\right)+\left(-\dfrac{85}{2}\right)\sqrt{5}+\left(\dfrac{152}{3}\right)\sqrt{5}+\left(24\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-\dfrac{215}{7}\right)\sqrt{5}+\left(-4\right)\sqrt{5}-\dfrac{34}{9}+\left(\left(-\dfrac{46}{7}\right)\sqrt{5}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{128}{5}\right)\sqrt{5}\right)-\dfrac{76}{7}-\left(\left(\dfrac{42}{5}\right)\sqrt{5}\right)+\left(-\dfrac{75}{8}\right)\sqrt{5}+\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(13\right)\sqrt{5}\right)+\left(\left(-18\right)\sqrt{5}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{5}\right)-\dfrac{3}{5}\\ &=&\dfrac{102289}{1260}+\left(-\dfrac{645}{8}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{75}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{20}\right)+\left(-\dfrac{17}{2}\right)\sqrt{125}+\left(\dfrac{76}{3}\right)\sqrt{20}+\left(8\right)\sqrt{45}+\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{125}+\left(-2\right)\sqrt{20}-\dfrac{34}{9}\right)-\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{20}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{20}\right)-\dfrac{76}{7}-\left(\left(\dfrac{21}{5}\right)\sqrt{20}\right)+\left(-\dfrac{15}{8}\right)\sqrt{125}+\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{13}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{4}{7}\right)\sqrt{45}\right)-\dfrac{3}{5}\right)\\ &=&\left(\dfrac{375}{4}-\left(\left(-4\right)\sqrt{5}\right)+\left(-\dfrac{85}{2}\right)\sqrt{5}+\left(\dfrac{152}{3}\right)\sqrt{5}+\left(24\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-\dfrac{215}{7}\right)\sqrt{5}+\left(-4\right)\sqrt{5}-\dfrac{34}{9}\right)-\left(\left(\left(-\dfrac{46}{7}\right)\sqrt{5}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{128}{5}\right)\sqrt{5}\right)-\dfrac{76}{7}-\left(\left(\dfrac{42}{5}\right)\sqrt{5}\right)+\left(-\dfrac{75}{8}\right)\sqrt{5}+\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(13\right)\sqrt{5}\right)+\left(\left(-18\right)\sqrt{5}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{5}\right)-\dfrac{3}{5}\right)\\ &=&\left(\dfrac{3239}{36}+\left(\dfrac{1571}{84}\right)\sqrt{5}\right)-\left(\left(-\dfrac{16687}{168}\right)\sqrt{5}-\dfrac{923}{105}\right)\\ &=&\dfrac{3239}{36}+\left(\dfrac{1571}{84}\right)\sqrt{5}+\left(\dfrac{16687}{168}\right)\sqrt{5}+\dfrac{923}{105}\\ &=&\dfrac{124441}{1260}+\left(\dfrac{19829}{168}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{75}{4}\right)\sqrt{25}\right)-\left(\left(-2\right)\sqrt{20}\right)+\left(-\dfrac{17}{2}\right)\sqrt{125}+\left(\dfrac{76}{3}\right)\sqrt{20}+\left(8\right)\sqrt{45}+\left(\dfrac{23}{4}\right)\sqrt{45}+\left(-\dfrac{43}{7}\right)\sqrt{125}+\left(-2\right)\sqrt{20}-\dfrac{34}{9}\right)\times\left(\left(\left(-\dfrac{23}{7}\right)\sqrt{20}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{38}{3}\right)\sqrt{125}\right)-\left(\left(\dfrac{64}{5}\right)\sqrt{20}\right)-\dfrac{76}{7}-\left(\left(\dfrac{21}{5}\right)\sqrt{20}\right)+\left(-\dfrac{15}{8}\right)\sqrt{125}+\left(\left(\dfrac{70}{3}\right)\sqrt{20}\right)-\left(\left(\dfrac{13}{5}\right)\sqrt{125}\right)+\left(\left(-\dfrac{18}{5}\right)\sqrt{125}\right)-\left(\left(\dfrac{4}{7}\right)\sqrt{45}\right)-\dfrac{3}{5}\right)\\ &=&\left(\dfrac{375}{4}-\left(\left(-4\right)\sqrt{5}\right)+\left(-\dfrac{85}{2}\right)\sqrt{5}+\left(\dfrac{152}{3}\right)\sqrt{5}+\left(24\right)\sqrt{5}+\left(\dfrac{69}{4}\right)\sqrt{5}+\left(-\dfrac{215}{7}\right)\sqrt{5}+\left(-4\right)\sqrt{5}-\dfrac{34}{9}\right)\times\left(\left(\left(-\dfrac{46}{7}\right)\sqrt{5}\right)+9-\dfrac{19}{3}-\left(\left(\dfrac{190}{3}\right)\sqrt{5}\right)-\left(\left(\dfrac{128}{5}\right)\sqrt{5}\right)-\dfrac{76}{7}-\left(\left(\dfrac{42}{5}\right)\sqrt{5}\right)+\left(-\dfrac{75}{8}\right)\sqrt{5}+\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(13\right)\sqrt{5}\right)+\left(\left(-18\right)\sqrt{5}\right)-\left(\left(\dfrac{12}{7}\right)\sqrt{5}\right)-\dfrac{3}{5}\right)\\ &=&\left(\dfrac{3239}{36}+\left(\dfrac{1571}{84}\right)\sqrt{5}\right)\left(\left(-\dfrac{16687}{168}\right)\sqrt{5}-\dfrac{923}{105}\right)\\ &=&\left(-\dfrac{485483681844}{53343360}\right)\sqrt{5}-\dfrac{2989597}{3780}+\left(-\dfrac{26215277}{14112}\right)\sqrt{25}\\ &=&\left(-\dfrac{485483681844}{53343360}\right)\sqrt{5}-\dfrac{537657928164}{53343360}\\ \end{eqnarray*}