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{26}{7}\right)\sqrt{50}\right)+1-\left(\left(-\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(\dfrac{51}{4}\right)\sqrt{8}+\left(-\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{47}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{8}\right)-\dfrac{32}{7}\right)-\left(\left(-5\right)\sqrt{8}+\left(-\dfrac{70}{3}\right)\sqrt{18}+\left(-\dfrac{80}{3}\right)\sqrt{8}\right)$ et $ Y=\left(\left(\dfrac{59}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{45}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{18}+\left(\dfrac{45}{2}\right)\sqrt{50}+\dfrac{19}{3}\right)$ . 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{26}{7}\right)\sqrt{50}\right)+1-\left(\left(-\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(\dfrac{51}{4}\right)\sqrt{8}+\left(-\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{47}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{8}\right)-\dfrac{32}{7}\right)-\left(\left(-5\right)\sqrt{8}+\left(-\dfrac{70}{3}\right)\sqrt{18}+\left(-\dfrac{80}{3}\right)\sqrt{8}\right)\right)+\left(\left(\left(\dfrac{59}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{45}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{18}+\left(\dfrac{45}{2}\right)\sqrt{50}+\dfrac{19}{3}\right)\right)\\ &=&\left(\left(\left(\left(\dfrac{130}{7}\right)\sqrt{2}\right)+1-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-15\right)\sqrt{2}+\left(\dfrac{51}{2}\right)\sqrt{2}-\dfrac{94}{7}\right)-\left(-\dfrac{94}{9}-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)-\left(\left(-10\right)\sqrt{2}+\left(-70\right)\sqrt{2}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)\right)+\left(\left(\dfrac{118}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{72}{5}\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}+\dfrac{19}{3}\right)\right)\\ &=&\left(\left(\left(\dfrac{130}{7}\right)\sqrt{2}\right)+1-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-15\right)\sqrt{2}+\left(\dfrac{51}{2}\right)\sqrt{2}-\dfrac{94}{7}\right)-\left(-\dfrac{94}{9}-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)-\left(\left(-10\right)\sqrt{2}+\left(-70\right)\sqrt{2}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)+\left(\dfrac{118}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{72}{5}\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}+\dfrac{19}{3}\right)\\ &=&\left(\dfrac{55127}{420}\right)\sqrt{2}+\dfrac{2518}{63}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\left(\dfrac{26}{7}\right)\sqrt{50}\right)+1-\left(\left(-\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(\dfrac{51}{4}\right)\sqrt{8}+\left(-\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{47}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{8}\right)-\dfrac{32}{7}\right)-\left(\left(-5\right)\sqrt{8}+\left(-\dfrac{70}{3}\right)\sqrt{18}+\left(-\dfrac{80}{3}\right)\sqrt{8}\right)\right)-\left(\left(\left(\dfrac{59}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{45}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{18}+\left(\dfrac{45}{2}\right)\sqrt{50}+\dfrac{19}{3}\right)\right)\\ &=&\left(\left(\left(\left(\dfrac{130}{7}\right)\sqrt{2}\right)+1-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-15\right)\sqrt{2}+\left(\dfrac{51}{2}\right)\sqrt{2}-\dfrac{94}{7}\right)-\left(-\dfrac{94}{9}-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)-\left(\left(-10\right)\sqrt{2}+\left(-70\right)\sqrt{2}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)\right)-\left(\left(\dfrac{118}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{72}{5}\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}+\dfrac{19}{3}\right)\right)\\ &=&\left(\left(\dfrac{3610}{21}\right)\sqrt{2}+\dfrac{265}{9}\right)-\left(\dfrac{221}{21}+\left(-\dfrac{813}{20}\right)\sqrt{2}\right)\\ &=&\left(\dfrac{3610}{21}\right)\sqrt{2}+\dfrac{265}{9}+-\dfrac{221}{21}+\left(\dfrac{813}{20}\right)\sqrt{2}\\ &=&\left(\dfrac{89273}{420}\right)\sqrt{2}+\dfrac{1192}{63}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\left(\dfrac{26}{7}\right)\sqrt{50}\right)+1-\left(\left(-\dfrac{5}{2}\right)\sqrt{8}\right)\right)-\left(\left(-3\right)\sqrt{50}+\left(\dfrac{51}{4}\right)\sqrt{8}+\left(-\dfrac{47}{7}\right)\sqrt{4}\right)-\left(\left(\left(-\dfrac{47}{9}\right)\sqrt{4}\right)-\left(\left(\dfrac{51}{4}\right)\sqrt{8}\right)-\dfrac{32}{7}\right)-\left(\left(-5\right)\sqrt{8}+\left(-\dfrac{70}{3}\right)\sqrt{18}+\left(-\dfrac{80}{3}\right)\sqrt{8}\right)\right)\times\left(\left(\left(\dfrac{59}{7}\right)\sqrt{4}+\left(1\right)\sqrt{50}+\left(\dfrac{45}{2}\right)\sqrt{50}\right)-\left(\left(\dfrac{25}{4}\right)\sqrt{50}\right)-\left(\left(\dfrac{24}{5}\right)\sqrt{18}+\left(\dfrac{45}{2}\right)\sqrt{50}+\dfrac{19}{3}\right)\right)\\ &=&\left(\left(\left(\left(\dfrac{130}{7}\right)\sqrt{2}\right)+1-\left(\left(-5\right)\sqrt{2}\right)\right)-\left(\left(-15\right)\sqrt{2}+\left(\dfrac{51}{2}\right)\sqrt{2}-\dfrac{94}{7}\right)-\left(-\dfrac{94}{9}-\left(\left(\dfrac{51}{2}\right)\sqrt{2}\right)-\dfrac{32}{7}\right)-\left(\left(-10\right)\sqrt{2}+\left(-70\right)\sqrt{2}+\left(-\dfrac{160}{3}\right)\sqrt{2}\right)\right)\times\left(\left(\dfrac{118}{7}+\left(5\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}\right)-\left(\left(\dfrac{125}{4}\right)\sqrt{2}\right)-\left(\left(\dfrac{72}{5}\right)\sqrt{2}+\left(\dfrac{225}{2}\right)\sqrt{2}+\dfrac{19}{3}\right)\right)\\ &=&\left(\left(\dfrac{3610}{21}\right)\sqrt{2}+\dfrac{265}{9}\right)\left(\dfrac{221}{21}+\left(-\dfrac{813}{20}\right)\sqrt{2}\right)\\ &=&\left(\dfrac{1079879}{1764}\right)\sqrt{2}+\left(-\dfrac{97831}{14}\right)\sqrt{4}+\dfrac{58565}{189}\\ &=&\left(\dfrac{1079879}{1764}\right)\sqrt{2}-\dfrac{2582872}{189}\\ \end{eqnarray*}