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(-\dfrac{26}{9}\right)\sqrt{45}+\left(\dfrac{76}{3}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{31}{2}\right)\sqrt{125}-\dfrac{23}{2}+\left(-\dfrac{68}{9}\right)\sqrt{45}+\left(\left(-\dfrac{23}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}\right)+\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)-\left(\left(8\right)\sqrt{45}\right)+\dfrac{23}{4}-\left(\left(3\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{125}\right)-\dfrac{68}{3}$ et $ Y=\left(\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{125}-8-\dfrac{75}{2}+\left(-7\right)\sqrt{125}+\left(-\dfrac{15}{2}\right)\sqrt{125}+\left(9\right)\sqrt{20}+\left(8\right)\sqrt{125}+\left(-\dfrac{49}{4}\right)\sqrt{45}+\dfrac{21}{8}$ . Calculer et simplifier $ X+Y$ , $ X-Y$ et $ X\times Y$ .

Cliquer ici pour afficher la solution

Exercice

\begin{eqnarray*} X+Y &=&\left(\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(\dfrac{76}{3}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{31}{2}\right)\sqrt{125}-\dfrac{23}{2}+\left(-\dfrac{68}{9}\right)\sqrt{45}+\left(\left(-\dfrac{23}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}\right)+\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)-\left(\left(8\right)\sqrt{45}\right)+\dfrac{23}{4}-\left(\left(3\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{125}\right)-\dfrac{68}{3}\right)+\left(\left(\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{125}-8-\dfrac{75}{2}+\left(-7\right)\sqrt{125}+\left(-\dfrac{15}{2}\right)\sqrt{125}+\left(9\right)\sqrt{20}+\left(8\right)\sqrt{125}+\left(-\dfrac{49}{4}\right)\sqrt{45}+\dfrac{21}{8}\right)\\ &=&\left(\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(\dfrac{380}{3}\right)\sqrt{5}-\dfrac{115}{3}+\left(-\dfrac{155}{2}\right)\sqrt{5}-\dfrac{23}{2}+\left(-\dfrac{68}{3}\right)\sqrt{5}-\dfrac{115}{2}+\dfrac{115}{3}+\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)+\dfrac{23}{4}-15-\left(\left(-20\right)\sqrt{5}\right)-\dfrac{68}{3}\right)+\left(\left(\dfrac{104}{9}\right)\sqrt{5}+\left(40\right)\sqrt{5}-8-\dfrac{75}{2}+\left(-35\right)\sqrt{5}+\left(-\dfrac{75}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(40\right)\sqrt{5}+\left(-\dfrac{147}{4}\right)\sqrt{5}+\dfrac{21}{8}\right)\\ &=&\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(\dfrac{380}{3}\right)\sqrt{5}-\dfrac{115}{3}+\left(-\dfrac{155}{2}\right)\sqrt{5}-\dfrac{23}{2}+\left(-\dfrac{68}{3}\right)\sqrt{5}-\dfrac{115}{2}+\dfrac{115}{3}+\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)+\dfrac{23}{4}-15-\left(\left(-20\right)\sqrt{5}\right)-\dfrac{68}{3}+\left(\dfrac{104}{9}\right)\sqrt{5}+\left(40\right)\sqrt{5}-8-\dfrac{75}{2}+\left(-35\right)\sqrt{5}+\left(-\dfrac{75}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(40\right)\sqrt{5}+\left(-\dfrac{147}{4}\right)\sqrt{5}+\dfrac{21}{8}\\ &=&\left(-\dfrac{739}{36}\right)\sqrt{5}-\dfrac{3451}{24}\\ \end{eqnarray*} \begin{eqnarray*} X-Y &=&\left(\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(\dfrac{76}{3}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{31}{2}\right)\sqrt{125}-\dfrac{23}{2}+\left(-\dfrac{68}{9}\right)\sqrt{45}+\left(\left(-\dfrac{23}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}\right)+\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)-\left(\left(8\right)\sqrt{45}\right)+\dfrac{23}{4}-\left(\left(3\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{125}\right)-\dfrac{68}{3}\right)-\left(\left(\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{125}-8-\dfrac{75}{2}+\left(-7\right)\sqrt{125}+\left(-\dfrac{15}{2}\right)\sqrt{125}+\left(9\right)\sqrt{20}+\left(8\right)\sqrt{125}+\left(-\dfrac{49}{4}\right)\sqrt{45}+\dfrac{21}{8}\right)\\ &=&\left(\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(\dfrac{380}{3}\right)\sqrt{5}-\dfrac{115}{3}+\left(-\dfrac{155}{2}\right)\sqrt{5}-\dfrac{23}{2}+\left(-\dfrac{68}{3}\right)\sqrt{5}-\dfrac{115}{2}+\dfrac{115}{3}+\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)+\dfrac{23}{4}-15-\left(\left(-20\right)\sqrt{5}\right)-\dfrac{68}{3}\right)-\left(\left(\dfrac{104}{9}\right)\sqrt{5}+\left(40\right)\sqrt{5}-8-\dfrac{75}{2}+\left(-35\right)\sqrt{5}+\left(-\dfrac{75}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(40\right)\sqrt{5}+\left(-\dfrac{147}{4}\right)\sqrt{5}+\dfrac{21}{8}\right)\\ &=&\left(\left(-\dfrac{125}{6}\right)\sqrt{5}-\dfrac{1211}{12}\right)-\left(\left(\dfrac{11}{36}\right)\sqrt{5}-\dfrac{343}{8}\right)\\ &=&\left(-\dfrac{125}{6}\right)\sqrt{5}-\dfrac{1211}{12}+\left(-\dfrac{11}{36}\right)\sqrt{5}+\dfrac{343}{8}\\ &=&\left(-\dfrac{761}{36}\right)\sqrt{5}-\dfrac{1393}{24}\\ \end{eqnarray*} \begin{eqnarray*} X\times Y &=&\left(\left(-\dfrac{26}{9}\right)\sqrt{45}+\left(\dfrac{76}{3}\right)\sqrt{125}+\left(-\dfrac{23}{3}\right)\sqrt{25}+\left(-\dfrac{31}{2}\right)\sqrt{125}-\dfrac{23}{2}+\left(-\dfrac{68}{9}\right)\sqrt{45}+\left(\left(-\dfrac{23}{2}\right)\sqrt{25}\right)-\left(\left(-\dfrac{23}{3}\right)\sqrt{25}\right)+\left(\left(-\dfrac{52}{3}\right)\sqrt{20}\right)-\left(\left(8\right)\sqrt{45}\right)+\dfrac{23}{4}-\left(\left(3\right)\sqrt{25}\right)-\left(\left(-4\right)\sqrt{125}\right)-\dfrac{68}{3}\right)\times\left(\left(\dfrac{52}{9}\right)\sqrt{20}+\left(8\right)\sqrt{125}-8-\dfrac{75}{2}+\left(-7\right)\sqrt{125}+\left(-\dfrac{15}{2}\right)\sqrt{125}+\left(9\right)\sqrt{20}+\left(8\right)\sqrt{125}+\left(-\dfrac{49}{4}\right)\sqrt{45}+\dfrac{21}{8}\right)\\ &=&\left(\left(-\dfrac{26}{3}\right)\sqrt{5}+\left(\dfrac{380}{3}\right)\sqrt{5}-\dfrac{115}{3}+\left(-\dfrac{155}{2}\right)\sqrt{5}-\dfrac{23}{2}+\left(-\dfrac{68}{3}\right)\sqrt{5}-\dfrac{115}{2}+\dfrac{115}{3}+\left(\left(-\dfrac{104}{3}\right)\sqrt{5}\right)-\left(\left(24\right)\sqrt{5}\right)+\dfrac{23}{4}-15-\left(\left(-20\right)\sqrt{5}\right)-\dfrac{68}{3}\right)\times\left(\left(\dfrac{104}{9}\right)\sqrt{5}+\left(40\right)\sqrt{5}-8-\dfrac{75}{2}+\left(-35\right)\sqrt{5}+\left(-\dfrac{75}{2}\right)\sqrt{5}+\left(18\right)\sqrt{5}+\left(40\right)\sqrt{5}+\left(-\dfrac{147}{4}\right)\sqrt{5}+\dfrac{21}{8}\right)\\ &=&\left(\left(-\dfrac{125}{6}\right)\sqrt{5}-\dfrac{1211}{12}\right)\left(\left(\dfrac{11}{36}\right)\sqrt{5}-\dfrac{343}{8}\right)\\ &=&\left(-\dfrac{1375}{216}\right)\sqrt{25}+\left(\dfrac{186277}{216}\right)\sqrt{5}+\dfrac{415373}{96}\\ &=&\dfrac{3710857}{864}+\left(\dfrac{186277}{216}\right)\sqrt{5}\\ \end{eqnarray*}