Prove that ((2+(√5)))^(1/3) +((2−(√5)))^(1/3) is a rational number


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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

	$L e t x = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$ $W e t h e n h a v e x - \sqrt[3]{2 + \sqrt{5}} - \sqrt[3]{2 - \sqrt{5}} = 0$ $a s w e h a v e s e e n a + b + c = 0 i m p l i e s$ $a^{3} + b^{3} + c^{3} = 3 a b c s o w e o b t a i n$ $x^{3} - (2 + \sqrt{5}) - (2 - \sqrt{5}) = 3 x \sqrt[3]{(2 + \sqrt{5}) (2 - \sqrt{5})}$ $o r x^{3} + 3 x - 4 = 0 . c l e a r l y o n e o f$ $t h e r o o t s o f t h i s e q u a t i o n i s x = 1$ $a n d t h e o t h e r t w o r o o t s s a t i s f y$ $t h e e q u a t i o n x^{2} + x + 4 = 0 w h i c h$ $h a s n o r e a l s o l u t i o n s . s i n c e$ $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} i s a r e a l r o o t$ $i t f o l l o w s t h a t \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} = 1$ $w h i c h a r a t i o n a l n u m b e r$
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