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{9}{2}\right)\sqrt{8}+\left(\dfrac{34}{5}\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{27}{5}\right)\sqrt{4}+\left(-4\right)\sqrt{4}+\left(-\dfrac{39}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{18}\right)-\left(\left(5\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{8}+\left(-2\right)\sqrt{18}+\left(\dfrac{19}{2}\right)\sqrt{4}\right)$ et $ Y=\left(-\dfrac{39}{2}\right)\sqrt{50}-\dfrac{47}{9}+\left(0\right)\sqrt{50}-\dfrac{3}{8}-\left(\left(-6\right)\sqrt{18}\right)-\left(\left(-4\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{18}\right)+\left(-\dfrac{51}{4}\right)\sqrt{18}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{34}{5}\right)\sqrt{18}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{19}{9}\right)\sqrt{50}-\dfrac{28}{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(-\dfrac{9}{2}\right)\sqrt{8}+\left(\dfrac{34}{5}\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{27}{5}\right)\sqrt{4}+\left(-4\right)\sqrt{4}+\left(-\dfrac{39}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{18}\right)-\left(\left(5\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{8}+\left(-2\right)\sqrt{18}+\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)+\left(\left(-\dfrac{39}{2}\right)\sqrt{50}-\dfrac{47}{9}+\left(0\right)\sqrt{50}-\dfrac{3}{8}-\left(\left(-6\right)\sqrt{18}\right)-\left(\left(-4\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{18}\right)+\left(-\dfrac{51}{4}\right)\sqrt{18}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{34}{5}\right)\sqrt{18}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{19}{9}\right)\sqrt{50}-\dfrac{28}{3}\right)\\ &=&\left(\left(\left(-9\right)\sqrt{2}+\left(\dfrac{102}{5}\right)\sqrt{2}+\dfrac{22}{3}\right)-\left(-33+\dfrac{54}{5}-8-\dfrac{78}{7}\right)-\left(\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}+\left(-\dfrac{158}{7}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+19\right)\right)+\left(\left(-\dfrac{195}{2}\right)\sqrt{2}-\dfrac{47}{9}+\left(0\right)\sqrt{2}-\dfrac{3}{8}-\left(\left(-18\right)\sqrt{2}\right)+8-\left(\left(-\dfrac{153}{4}\right)\sqrt{2}\right)+\left(-\dfrac{153}{4}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{102}{5}\right)\sqrt{2}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{95}{9}\right)\sqrt{2}-\dfrac{28}{3}\right)\\ &=&\left(\left(-9\right)\sqrt{2}+\left(\dfrac{102}{5}\right)\sqrt{2}+\dfrac{22}{3}\right)-\left(-33+\dfrac{54}{5}-8-\dfrac{78}{7}\right)-\left(\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}+\left(-\dfrac{158}{7}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+19\right)+\left(-\dfrac{195}{2}\right)\sqrt{2}-\dfrac{47}{9}+\left(0\right)\sqrt{2}-\dfrac{3}{8}-\left(\left(-18\right)\sqrt{2}\right)+8-\left(\left(-\dfrac{153}{4}\right)\sqrt{2}\right)+\left(-\dfrac{153}{4}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{102}{5}\right)\sqrt{2}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{95}{9}\right)\sqrt{2}-\dfrac{28}{3}\\ &=&\left(-\dfrac{59707}{630}\right)\sqrt{2}+\dfrac{2183}{140}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{8}+\left(\dfrac{34}{5}\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{27}{5}\right)\sqrt{4}+\left(-4\right)\sqrt{4}+\left(-\dfrac{39}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{18}\right)-\left(\left(5\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{8}+\left(-2\right)\sqrt{18}+\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)-\left(\left(-\dfrac{39}{2}\right)\sqrt{50}-\dfrac{47}{9}+\left(0\right)\sqrt{50}-\dfrac{3}{8}-\left(\left(-6\right)\sqrt{18}\right)-\left(\left(-4\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{18}\right)+\left(-\dfrac{51}{4}\right)\sqrt{18}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{34}{5}\right)\sqrt{18}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{19}{9}\right)\sqrt{50}-\dfrac{28}{3}\right)\\ &=&\left(\left(\left(-9\right)\sqrt{2}+\left(\dfrac{102}{5}\right)\sqrt{2}+\dfrac{22}{3}\right)-\left(-33+\dfrac{54}{5}-8-\dfrac{78}{7}\right)-\left(\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}+\left(-\dfrac{158}{7}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+19\right)\right)-\left(\left(-\dfrac{195}{2}\right)\sqrt{2}-\dfrac{47}{9}+\left(0\right)\sqrt{2}-\dfrac{3}{8}-\left(\left(-18\right)\sqrt{2}\right)+8-\left(\left(-\dfrac{153}{4}\right)\sqrt{2}\right)+\left(-\dfrac{153}{4}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{102}{5}\right)\sqrt{2}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{95}{9}\right)\sqrt{2}-\dfrac{28}{3}\right)\\ &=&\left(\left(-\dfrac{38}{7}\right)\sqrt{2}+\dfrac{3116}{105}\right)-\left(\left(-\dfrac{8041}{90}\right)\sqrt{2}-\dfrac{169}{12}\right)\\ &=&\left(-\dfrac{38}{7}\right)\sqrt{2}+\dfrac{3116}{105}+\left(\dfrac{8041}{90}\right)\sqrt{2}+\dfrac{169}{12}\\ &=&\left(\dfrac{52867}{630}\right)\sqrt{2}+\dfrac{18379}{420}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(-\dfrac{9}{2}\right)\sqrt{8}+\left(\dfrac{34}{5}\right)\sqrt{18}+\left(\dfrac{11}{3}\right)\sqrt{4}\right)-\left(\left(-\dfrac{33}{2}\right)\sqrt{4}+\left(\dfrac{27}{5}\right)\sqrt{4}+\left(-4\right)\sqrt{4}+\left(-\dfrac{39}{7}\right)\sqrt{4}\right)-\left(\left(\dfrac{34}{5}\right)\sqrt{18}\right)-\left(\left(5\right)\sqrt{50}+\left(-\dfrac{79}{7}\right)\sqrt{8}+\left(-2\right)\sqrt{18}+\left(\dfrac{19}{2}\right)\sqrt{4}\right)\right)\times\left(\left(-\dfrac{39}{2}\right)\sqrt{50}-\dfrac{47}{9}+\left(0\right)\sqrt{50}-\dfrac{3}{8}-\left(\left(-6\right)\sqrt{18}\right)-\left(\left(-4\right)\sqrt{4}\right)-\left(\left(-\dfrac{51}{4}\right)\sqrt{18}\right)+\left(-\dfrac{51}{4}\right)\sqrt{18}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{34}{5}\right)\sqrt{18}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{19}{9}\right)\sqrt{50}-\dfrac{28}{3}\right)\\ &=&\left(\left(\left(-9\right)\sqrt{2}+\left(\dfrac{102}{5}\right)\sqrt{2}+\dfrac{22}{3}\right)-\left(-33+\dfrac{54}{5}-8-\dfrac{78}{7}\right)-\left(\left(\dfrac{102}{5}\right)\sqrt{2}\right)-\left(\left(25\right)\sqrt{2}+\left(-\dfrac{158}{7}\right)\sqrt{2}+\left(-6\right)\sqrt{2}+19\right)\right)\times\left(\left(-\dfrac{195}{2}\right)\sqrt{2}-\dfrac{47}{9}+\left(0\right)\sqrt{2}-\dfrac{3}{8}-\left(\left(-18\right)\sqrt{2}\right)+8-\left(\left(-\dfrac{153}{4}\right)\sqrt{2}\right)+\left(-\dfrac{153}{4}\right)\sqrt{2}-\dfrac{47}{9}+\dfrac{33}{8}+\left(-\dfrac{102}{5}\right)\sqrt{2}+\dfrac{58}{9}-\dfrac{25}{2}+\left(\dfrac{95}{9}\right)\sqrt{2}-\dfrac{28}{3}\right)\\ &=&\left(\left(-\dfrac{38}{7}\right)\sqrt{2}+\dfrac{3116}{105}\right)\left(\left(-\dfrac{8041}{90}\right)\sqrt{2}-\dfrac{169}{12}\right)\\ &=&\left(\dfrac{152779}{315}\right)\sqrt{4}+\left(-\dfrac{3476183}{1350}\right)\sqrt{2}-\dfrac{131651}{315}\\ &=&\dfrac{19323}{35}+\left(-\dfrac{3476183}{1350}\right)\sqrt{2}\\ \end{eqnarray*}