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{81}{5}\right)\sqrt{18}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{9}\right)\sqrt{18}\right)+\dfrac{46}{9}\right)-\left(\left(-7\right)\sqrt{8}-\dfrac{46}{9}+\left(0\right)\sqrt{4}-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{7}{2}\right)\sqrt{50}+\left(-\dfrac{68}{7}\right)\sqrt{8}+\left(-\dfrac{27}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{8}-\dfrac{68}{7}+0\right)$ et $ Y=\left(\dfrac{33}{7}\right)\sqrt{50}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{4}\right)+9-\left(\left(-9\right)\sqrt{8}\right)+\left(-9\right)\sqrt{50}+\dfrac{32}{7}+\left(\dfrac{33}{7}\right)\sqrt{50}-\dfrac{11}{2}+7+\left(\dfrac{61}{6}\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{4}$ . 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{81}{5}\right)\sqrt{18}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{9}\right)\sqrt{18}\right)+\dfrac{46}{9}\right)-\left(\left(-7\right)\sqrt{8}-\dfrac{46}{9}+\left(0\right)\sqrt{4}-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{7}{2}\right)\sqrt{50}+\left(-\dfrac{68}{7}\right)\sqrt{8}+\left(-\dfrac{27}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{8}-\dfrac{68}{7}+0\right)\right)+\left(\left(\dfrac{33}{7}\right)\sqrt{50}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{4}\right)+9-\left(\left(-9\right)\sqrt{8}\right)+\left(-9\right)\sqrt{50}+\dfrac{32}{7}+\left(\dfrac{33}{7}\right)\sqrt{50}-\dfrac{11}{2}+7+\left(\dfrac{61}{6}\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{4}\right)\\ &=&\left(\left(\left(\left(-\dfrac{243}{5}\right)\sqrt{2}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{3}\right)\sqrt{2}\right)+\dfrac{46}{9}\right)-\left(\left(-14\right)\sqrt{2}-\dfrac{46}{9}+0-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{136}{7}\right)\sqrt{2}+\left(-\dfrac{81}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{2}-\dfrac{68}{7}+0\right)\right)+\left(\left(\dfrac{165}{7}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)-\dfrac{110}{9}+9-\left(\left(-18\right)\sqrt{2}\right)+\left(-45\right)\sqrt{2}+\dfrac{32}{7}+\left(\dfrac{165}{7}\right)\sqrt{2}-\dfrac{11}{2}+7+\dfrac{61}{3}+27\right)\\ &=&\left(\left(\left(-\dfrac{243}{5}\right)\sqrt{2}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{3}\right)\sqrt{2}\right)+\dfrac{46}{9}\right)-\left(\left(-14\right)\sqrt{2}-\dfrac{46}{9}+0-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{136}{7}\right)\sqrt{2}+\left(-\dfrac{81}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{2}-\dfrac{68}{7}+0\right)+\left(\dfrac{165}{7}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)-\dfrac{110}{9}+9-\left(\left(-18\right)\sqrt{2}\right)+\left(-45\right)\sqrt{2}+\dfrac{32}{7}+\left(\dfrac{165}{7}\right)\sqrt{2}-\dfrac{11}{2}+7+\dfrac{61}{3}+27\\ &=&\left(\dfrac{3733}{420}\right)\sqrt{2}+\dfrac{21137}{210}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\left(-\dfrac{81}{5}\right)\sqrt{18}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{9}\right)\sqrt{18}\right)+\dfrac{46}{9}\right)-\left(\left(-7\right)\sqrt{8}-\dfrac{46}{9}+\left(0\right)\sqrt{4}-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{7}{2}\right)\sqrt{50}+\left(-\dfrac{68}{7}\right)\sqrt{8}+\left(-\dfrac{27}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{8}-\dfrac{68}{7}+0\right)\right)-\left(\left(\dfrac{33}{7}\right)\sqrt{50}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{4}\right)+9-\left(\left(-9\right)\sqrt{8}\right)+\left(-9\right)\sqrt{50}+\dfrac{32}{7}+\left(\dfrac{33}{7}\right)\sqrt{50}-\dfrac{11}{2}+7+\left(\dfrac{61}{6}\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{4}\right)\\ &=&\left(\left(\left(\left(-\dfrac{243}{5}\right)\sqrt{2}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{3}\right)\sqrt{2}\right)+\dfrac{46}{9}\right)-\left(\left(-14\right)\sqrt{2}-\dfrac{46}{9}+0-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{136}{7}\right)\sqrt{2}+\left(-\dfrac{81}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{2}-\dfrac{68}{7}+0\right)\right)-\left(\left(\dfrac{165}{7}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)-\dfrac{110}{9}+9-\left(\left(-18\right)\sqrt{2}\right)+\left(-45\right)\sqrt{2}+\dfrac{32}{7}+\left(\dfrac{165}{7}\right)\sqrt{2}-\dfrac{11}{2}+7+\dfrac{61}{3}+27\right)\\ &=&\left(\left(-\dfrac{4727}{420}\right)\sqrt{2}+\dfrac{15898}{315}\right)-\left(\left(\dfrac{141}{7}\right)\sqrt{2}+\dfrac{6323}{126}\right)\\ &=&\left(-\dfrac{4727}{420}\right)\sqrt{2}+\dfrac{15898}{315}+\left(-\dfrac{141}{7}\right)\sqrt{2}-\dfrac{6323}{126}\\ &=&\left(-\dfrac{13187}{420}\right)\sqrt{2}+\dfrac{181}{630}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\left(-\dfrac{81}{5}\right)\sqrt{18}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{9}\right)\sqrt{18}\right)+\dfrac{46}{9}\right)-\left(\left(-7\right)\sqrt{8}-\dfrac{46}{9}+\left(0\right)\sqrt{4}-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{7}{2}\right)\sqrt{50}+\left(-\dfrac{68}{7}\right)\sqrt{8}+\left(-\dfrac{27}{2}\right)\sqrt{18}\right)-\left(\left(-\dfrac{13}{8}\right)\sqrt{8}-\dfrac{68}{7}+0\right)\right)\times\left(\left(\dfrac{33}{7}\right)\sqrt{50}+\left(\left(0\right)\sqrt{8}\right)-\left(\left(\dfrac{55}{9}\right)\sqrt{4}\right)+9-\left(\left(-9\right)\sqrt{8}\right)+\left(-9\right)\sqrt{50}+\dfrac{32}{7}+\left(\dfrac{33}{7}\right)\sqrt{50}-\dfrac{11}{2}+7+\left(\dfrac{61}{6}\right)\sqrt{4}+\left(\dfrac{27}{2}\right)\sqrt{4}\right)\\ &=&\left(\left(\left(\left(-\dfrac{243}{5}\right)\sqrt{2}\right)-\dfrac{14}{5}-\left(\left(\dfrac{67}{3}\right)\sqrt{2}\right)+\dfrac{46}{9}\right)-\left(\left(-14\right)\sqrt{2}-\dfrac{46}{9}+0-\dfrac{43}{2}\right)-\left(-\dfrac{71}{6}+\left(\dfrac{35}{2}\right)\sqrt{2}+\left(-\dfrac{136}{7}\right)\sqrt{2}+\left(-\dfrac{81}{2}\right)\sqrt{2}\right)-\left(\left(-\dfrac{13}{4}\right)\sqrt{2}-\dfrac{68}{7}+0\right)\right)\times\left(\left(\dfrac{165}{7}\right)\sqrt{2}+\left(\left(0\right)\sqrt{2}\right)-\dfrac{110}{9}+9-\left(\left(-18\right)\sqrt{2}\right)+\left(-45\right)\sqrt{2}+\dfrac{32}{7}+\left(\dfrac{165}{7}\right)\sqrt{2}-\dfrac{11}{2}+7+\dfrac{61}{3}+27\right)\\ &=&\left(\left(-\dfrac{4727}{420}\right)\sqrt{2}+\dfrac{15898}{315}\right)\left(\left(\dfrac{141}{7}\right)\sqrt{2}+\dfrac{6323}{126}\right)\\ &=&\left(-\dfrac{222169}{980}\right)\sqrt{4}+\left(\dfrac{17573858085}{38896200}\right)\sqrt{2}+\dfrac{50261527}{19845}\\ &=&\dfrac{20219204425}{9724050}+\left(\dfrac{17573858085}{38896200}\right)\sqrt{2}\\ \end{eqnarray*}