Prove-that-2-5-1-3-2-5-1-3-is-a-rational-number-

Question Number 149883 by liberty last updated on 08/Aug/21

Prove that \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} is

a rational number

Answered by john_santu last updated on 08/Aug/21

L e t x = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}

W e t h e n h a v e x - \sqrt[3]{2 + \sqrt{5}} - \sqrt[3]{2 - \sqrt{5}} = 0

a s w e h a v e s e e n a + b + c = 0 i m p l i e s

a^{3} + b^{3} + c^{3} = 3 a b c s o w e o b t a i n

x^{3} - (2 + \sqrt{5}) - (2 - \sqrt{5}) = 3 x \sqrt[3]{(2 + \sqrt{5}) (2 - \sqrt{5})}

o r x^{3} + 3 x - 4 = 0 . c l e a r l y o n e o f

t h e r o o t s o f t h i s e q u a t i o n i s x = 1

a n d t h e o t h e r t w o r o o t s s a t i s f y

t h e e q u a t i o n x^{2} + x + 4 = 0 w h i c h

h a s n o r e a l s o l u t i o n s . s i n c e

\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} i s a r e a l r o o t

i t f o l l o w s t h a t \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} = 1

w h i c h a r a t i o n a l n u m b e r