Question

13 Calculaţi x = 30 a + b şi y=a: b, unde √3-√√√5 √5-√2 2 3 b = √√√2³ -√3³ + √5³ a √√2-√√3 5 +
ajutor vă rog ​

Answer 1

Explicație pas cu pas:

$a = \dfrac{ \sqrt{2} - \sqrt{3} }{5} + \dfrac{ \sqrt{3} - \sqrt{5} }{2} + \dfrac{ \sqrt{5} - \sqrt{2} }{3} = \\ = \dfrac{ 6(\sqrt{2} - \sqrt{3}) + 15(\sqrt{3} - \sqrt{5}) + 10(\sqrt{5} - \sqrt{2})}{30} \\ = \dfrac{ 6\sqrt{2} - 6\sqrt{3} + 15\sqrt{3} - 15\sqrt{5} + 10\sqrt{5} - 10\sqrt{2}}{30}$

$a = \dfrac{- 4\sqrt{2} + 9\sqrt{3} - 5\sqrt{5}}{30}$

$b = { \sqrt{2} }^{5} - { \sqrt{3} }^{5} + { \sqrt{5} }^{3} = {2}^{2} \sqrt{2} - {3}^{2} \sqrt{3} + 5 \sqrt{5}$

$b = 4 \sqrt{2} - 9 \sqrt{3} + 5 \sqrt{5}$

=>

$x = 30 \cdot a + b = 30 \cdot \dfrac{- 4\sqrt{2} + 9\sqrt{3} - 5\sqrt{5}}{30} + 4 \sqrt{2} - 9 \sqrt{3} + 5 \sqrt{5} = - 4\sqrt{2} + 9\sqrt{3} - 5\sqrt{5} + 4 \sqrt{2} - 9 \sqrt{3} + 5 \sqrt{5} = 0$

$\implies \bf x = 0$

$y = a : b = -\dfrac{4\sqrt{2} - 9\sqrt{3} + 5\sqrt{5}}{30} : (4 \sqrt{2} - 9 \sqrt{3} + 5 \sqrt{5}) = -\dfrac{4\sqrt{2} - 9\sqrt{3} + 5\sqrt{5}}{30 \cdot (4\sqrt{2} - 9\sqrt{3} + 5\sqrt{5})} = - \dfrac{1}{30}$

$\implies \bf y = - \dfrac{1}{30}$