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{55}{3}\right)\sqrt{125}+\left(-\dfrac{13}{4}\right)\sqrt{45}+\left(-\dfrac{75}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{28}{9}\right)\sqrt{125}-4+\left(\dfrac{5}{3}\right)\sqrt{20}-\dfrac{38}{3}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{25}{2}\right)\sqrt{45}\right)+2-\left(\left(-\dfrac{5}{3}\right)\sqrt{20}\right)\right)$ et $ Y=\left(-8\right)\sqrt{45}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(9\right)\sqrt{45}+\left(\dfrac{81}{8}\right)\sqrt{45}-\dfrac{13}{5}-\left(\left(\dfrac{13}{3}\right)\sqrt{45}\right)+\left(-\dfrac{29}{2}\right)\sqrt{20}+\left(\dfrac{13}{3}\right)\sqrt{45}+\left(-8\right)\sqrt{125}+\left(-\dfrac{23}{2}\right)\sqrt{20}$ . 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{55}{3}\right)\sqrt{125}+\left(-\dfrac{13}{4}\right)\sqrt{45}+\left(-\dfrac{75}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{28}{9}\right)\sqrt{125}-4+\left(\dfrac{5}{3}\right)\sqrt{20}-\dfrac{38}{3}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{25}{2}\right)\sqrt{45}\right)+2-\left(\left(-\dfrac{5}{3}\right)\sqrt{20}\right)\right)\right)+\left(\left(-8\right)\sqrt{45}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(9\right)\sqrt{45}+\left(\dfrac{81}{8}\right)\sqrt{45}-\dfrac{13}{5}-\left(\left(\dfrac{13}{3}\right)\sqrt{45}\right)+\left(-\dfrac{29}{2}\right)\sqrt{20}+\left(\dfrac{13}{3}\right)\sqrt{45}+\left(-8\right)\sqrt{125}+\left(-\dfrac{23}{2}\right)\sqrt{20}\right)\\ &=&\left(\left(\left(\dfrac{275}{3}\right)\sqrt{5}+\left(-\dfrac{39}{4}\right)\sqrt{5}+\left(-\dfrac{375}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{9}\right)\sqrt{5}-4+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{38}{3}\right)-\left(-\dfrac{45}{2}-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{145}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{5}\right)+2-\left(\left(-\dfrac{10}{3}\right)\sqrt{5}\right)\right)\right)+\left(\left(-24\right)\sqrt{5}+\dfrac{50}{3}+\left(27\right)\sqrt{5}+\left(\dfrac{243}{8}\right)\sqrt{5}-\dfrac{13}{5}-\left(\left(13\right)\sqrt{5}\right)+\left(-29\right)\sqrt{5}+\left(13\right)\sqrt{5}+\left(-40\right)\sqrt{5}+\left(-23\right)\sqrt{5}\right)\\ &=&\left(\left(\dfrac{275}{3}\right)\sqrt{5}+\left(-\dfrac{39}{4}\right)\sqrt{5}+\left(-\dfrac{375}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{9}\right)\sqrt{5}-4+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{38}{3}\right)-\left(-\dfrac{45}{2}-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{145}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{5}\right)+2-\left(\left(-\dfrac{10}{3}\right)\sqrt{5}\right)\right)+\left(-24\right)\sqrt{5}+\dfrac{50}{3}+\left(27\right)\sqrt{5}+\left(\dfrac{243}{8}\right)\sqrt{5}-\dfrac{13}{5}-\left(\left(13\right)\sqrt{5}\right)+\left(-29\right)\sqrt{5}+\left(13\right)\sqrt{5}+\left(-40\right)\sqrt{5}+\left(-23\right)\sqrt{5}\\ &=&\left(-\dfrac{14935}{72}\right)\sqrt{5}+\dfrac{4109}{60}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{55}{3}\right)\sqrt{125}+\left(-\dfrac{13}{4}\right)\sqrt{45}+\left(-\dfrac{75}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{28}{9}\right)\sqrt{125}-4+\left(\dfrac{5}{3}\right)\sqrt{20}-\dfrac{38}{3}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{25}{2}\right)\sqrt{45}\right)+2-\left(\left(-\dfrac{5}{3}\right)\sqrt{20}\right)\right)\right)-\left(\left(-8\right)\sqrt{45}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(9\right)\sqrt{45}+\left(\dfrac{81}{8}\right)\sqrt{45}-\dfrac{13}{5}-\left(\left(\dfrac{13}{3}\right)\sqrt{45}\right)+\left(-\dfrac{29}{2}\right)\sqrt{20}+\left(\dfrac{13}{3}\right)\sqrt{45}+\left(-8\right)\sqrt{125}+\left(-\dfrac{23}{2}\right)\sqrt{20}\right)\\ &=&\left(\left(\left(\dfrac{275}{3}\right)\sqrt{5}+\left(-\dfrac{39}{4}\right)\sqrt{5}+\left(-\dfrac{375}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{9}\right)\sqrt{5}-4+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{38}{3}\right)-\left(-\dfrac{45}{2}-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{145}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{5}\right)+2-\left(\left(-\dfrac{10}{3}\right)\sqrt{5}\right)\right)\right)-\left(\left(-24\right)\sqrt{5}+\dfrac{50}{3}+\left(27\right)\sqrt{5}+\left(\dfrac{243}{8}\right)\sqrt{5}-\dfrac{13}{5}-\left(\left(13\right)\sqrt{5}\right)+\left(-29\right)\sqrt{5}+\left(13\right)\sqrt{5}+\left(-40\right)\sqrt{5}+\left(-23\right)\sqrt{5}\right)\\ &=&\left(\left(-\dfrac{5357}{36}\right)\sqrt{5}+\dfrac{653}{12}\right)-\left(\left(-\dfrac{469}{8}\right)\sqrt{5}+\dfrac{211}{15}\right)\\ &=&\left(-\dfrac{5357}{36}\right)\sqrt{5}+\dfrac{653}{12}+\left(\dfrac{469}{8}\right)\sqrt{5}-\dfrac{211}{15}\\ &=&\left(-\dfrac{6493}{72}\right)\sqrt{5}+\dfrac{807}{20}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{55}{3}\right)\sqrt{125}+\left(-\dfrac{13}{4}\right)\sqrt{45}+\left(-\dfrac{75}{2}\right)\sqrt{125}\right)-\left(\left(\dfrac{28}{9}\right)\sqrt{125}-4+\left(\dfrac{5}{3}\right)\sqrt{20}-\dfrac{38}{3}\right)-\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{25}\right)-\left(\left(-3\right)\sqrt{20}\right)-\left(\left(\dfrac{28}{3}\right)\sqrt{125}\right)-\left(\left(-\dfrac{29}{6}\right)\sqrt{125}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{25}{2}\right)\sqrt{45}\right)+2-\left(\left(-\dfrac{5}{3}\right)\sqrt{20}\right)\right)\right)\times\left(\left(-8\right)\sqrt{45}+\left(\dfrac{10}{3}\right)\sqrt{25}+\left(9\right)\sqrt{45}+\left(\dfrac{81}{8}\right)\sqrt{45}-\dfrac{13}{5}-\left(\left(\dfrac{13}{3}\right)\sqrt{45}\right)+\left(-\dfrac{29}{2}\right)\sqrt{20}+\left(\dfrac{13}{3}\right)\sqrt{45}+\left(-8\right)\sqrt{125}+\left(-\dfrac{23}{2}\right)\sqrt{20}\right)\\ &=&\left(\left(\left(\dfrac{275}{3}\right)\sqrt{5}+\left(-\dfrac{39}{4}\right)\sqrt{5}+\left(-\dfrac{375}{2}\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{9}\right)\sqrt{5}-4+\left(\dfrac{10}{3}\right)\sqrt{5}-\dfrac{38}{3}\right)-\left(-\dfrac{45}{2}-\left(\left(-6\right)\sqrt{5}\right)-\left(\left(\dfrac{140}{3}\right)\sqrt{5}\right)-\left(\left(-\dfrac{145}{6}\right)\sqrt{5}\right)\right)-\left(-\dfrac{69}{4}-\left(\left(-\dfrac{75}{2}\right)\sqrt{5}\right)+2-\left(\left(-\dfrac{10}{3}\right)\sqrt{5}\right)\right)\right)\times\left(\left(-24\right)\sqrt{5}+\dfrac{50}{3}+\left(27\right)\sqrt{5}+\left(\dfrac{243}{8}\right)\sqrt{5}-\dfrac{13}{5}-\left(\left(13\right)\sqrt{5}\right)+\left(-29\right)\sqrt{5}+\left(13\right)\sqrt{5}+\left(-40\right)\sqrt{5}+\left(-23\right)\sqrt{5}\right)\\ &=&\left(\left(-\dfrac{5357}{36}\right)\sqrt{5}+\dfrac{653}{12}\right)\left(\left(-\dfrac{469}{8}\right)\sqrt{5}+\dfrac{211}{15}\right)\\ &=&\left(\dfrac{2512433}{288}\right)\sqrt{25}+\left(-\dfrac{22824181}{4320}\right)\sqrt{5}+\dfrac{137783}{180}\\ &=&\dfrac{2300871204}{51840}+\left(-\dfrac{22824181}{4320}\right)\sqrt{5}\\ \end{eqnarray*}