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{48}{5}\right)\sqrt{25}+\left(\dfrac{31}{4}\right)\sqrt{20}+\left(\dfrac{76}{9}\right)\sqrt{25}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{125}+\dfrac{19}{4}\right)$ et $ Y=\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)\right)-\left(4-\left(\left(-9\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(\left(2\right)\sqrt{20}\right)+4\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(\dfrac{48}{5}\right)\sqrt{25}+\left(\dfrac{31}{4}\right)\sqrt{20}+\left(\dfrac{76}{9}\right)\sqrt{25}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{125}+\dfrac{19}{4}\right)\right)+\left(\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)\right)-\left(4-\left(\left(-9\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(\left(2\right)\sqrt{20}\right)+4\right)\right)\\ &=&\left(\left(48+\left(\dfrac{31}{2}\right)\sqrt{5}+\dfrac{380}{9}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{98}{9}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{130}{3}\right)\sqrt{5}+\dfrac{19}{4}\right)\right)+\left(\left(\left(\left(-\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)\right)-\left(4-\left(\left(-18\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(\left(4\right)\sqrt{5}\right)+4\right)\right)\\ &=&\left(48+\left(\dfrac{31}{2}\right)\sqrt{5}+\dfrac{380}{9}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{98}{9}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{130}{3}\right)\sqrt{5}+\dfrac{19}{4}\right)+\left(\left(\left(-\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)\right)-\left(4-\left(\left(-18\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(\left(4\right)\sqrt{5}\right)+4\right)\\ &=&\dfrac{22171}{180}+\left(-\dfrac{4265}{36}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(\left(\dfrac{48}{5}\right)\sqrt{25}+\left(\dfrac{31}{4}\right)\sqrt{20}+\left(\dfrac{76}{9}\right)\sqrt{25}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{125}+\dfrac{19}{4}\right)\right)-\left(\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)\right)-\left(4-\left(\left(-9\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(\left(2\right)\sqrt{20}\right)+4\right)\right)\\ &=&\left(\left(48+\left(\dfrac{31}{2}\right)\sqrt{5}+\dfrac{380}{9}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{98}{9}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{130}{3}\right)\sqrt{5}+\dfrac{19}{4}\right)\right)-\left(\left(\left(\left(-\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)\right)-\left(4-\left(\left(-18\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(\left(4\right)\sqrt{5}\right)+4\right)\right)\\ &=&\left(\dfrac{23611}{180}+\left(-\dfrac{344}{9}\right)\sqrt{5}\right)-\left(\left(-\dfrac{321}{4}\right)\sqrt{5}-8\right)\\ &=&\dfrac{23611}{180}+\left(-\dfrac{344}{9}\right)\sqrt{5}+\left(\dfrac{321}{4}\right)\sqrt{5}+8\\ &=&\dfrac{25051}{180}+\left(\dfrac{1513}{36}\right)\sqrt{5}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(\left(\dfrac{48}{5}\right)\sqrt{25}+\left(\dfrac{31}{4}\right)\sqrt{20}+\left(\dfrac{76}{9}\right)\sqrt{25}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{4}\right)\sqrt{20}\right)-\left(\left(-\dfrac{49}{9}\right)\sqrt{20}\right)-\left(\left(4\right)\sqrt{20}\right)\right)-\left(\left(\dfrac{26}{3}\right)\sqrt{125}+\dfrac{19}{4}\right)\right)\times\left(\left(\left(\left(-\dfrac{1}{4}\right)\sqrt{125}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)\right)-\left(4-\left(\left(-9\right)\sqrt{20}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(-\dfrac{61}{3}\right)\sqrt{45}\right)-\left(\left(2\right)\sqrt{20}\right)\right)-\left(\left(\left(2\right)\sqrt{20}\right)+4\right)\right)\\ &=&\left(\left(48+\left(\dfrac{31}{2}\right)\sqrt{5}+\dfrac{380}{9}\right)-\left(-\dfrac{56}{5}-\dfrac{69}{2}\right)-\left(\left(\left(\dfrac{15}{2}\right)\sqrt{5}\right)-\left(\left(-\dfrac{98}{9}\right)\sqrt{5}\right)-\left(\left(8\right)\sqrt{5}\right)\right)-\left(\left(\dfrac{130}{3}\right)\sqrt{5}+\dfrac{19}{4}\right)\right)\times\left(\left(\left(\left(-\dfrac{5}{4}\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)\right)-\left(4-\left(\left(-18\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(-61\right)\sqrt{5}\right)-\left(\left(4\right)\sqrt{5}\right)\right)-\left(\left(\left(4\right)\sqrt{5}\right)+4\right)\right)\\ &=&\left(\dfrac{23611}{180}+\left(-\dfrac{344}{9}\right)\sqrt{5}\right)\left(\left(-\dfrac{321}{4}\right)\sqrt{5}-8\right)\\ &=&\left(-\dfrac{7358971}{720}\right)\sqrt{5}-\dfrac{47222}{45}+\left(\dfrac{9202}{3}\right)\sqrt{25}\\ &=&\left(-\dfrac{7358971}{720}\right)\sqrt{5}+\dfrac{642928}{45}\\ \end{eqnarray*}