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Question Number 46414 by Meritguide1234 last updated on 25/Oct/18

Commented by math1967 last updated on 25/Oct/18

$I s t h e a n s w a r 2 \sqrt{3}$
	Answered by math1967 last updated on 25/Oct/18

	$l e t \sqrt{3} + \sqrt{2} = a ∴ \sqrt{3} - \sqrt{2} = \frac{1}{a}$ $∴ x = \overset{5}{} \sqrt{a} + \frac{1}{\sqrt[5]{a}} = p + \frac{1}{p} [l e t \overset{5}{} \sqrt{a} = p]$ $x^{3} = p^{3} + \frac{1}{p^{3}} + 3 (p + \frac{1}{p})$ $x^{3} - 3 x = p^{3} + \frac{1}{p^{3}} [∵ p + \frac{1}{p} = x]$ $n o w x^{5} = p^{5} + \frac{1}{p^{5}} + 5 (p^{3} + \frac{1}{p^{3}}) + 10 (p + \frac{1}{p})$ $x^{5} = p^{5} + \frac{1}{p^{5}} + 5 (x^{3} - 3 x) + 10 x$ $x^{5} - 5 x^{3} + 5 x = \sqrt{3} + \sqrt{2} + \frac{1}{\sqrt{3} - \sqrt{2}}$ $∴ x^{3} - 5 x^{3} + 5 x = \sqrt{3} + \sqrt{2} + \sqrt{3} - \sqrt{2} = 2 \sqrt{3}$
	Commented by Meritguide1234 last updated on 26/Oct/18

	$n i c e s o l u t i o n$
	Commented by math1967 last updated on 26/Oct/18

	$T h a n k y o u s i r$
	Answered by tanmay.chaudhury50@gmail.com last updated on 25/Oct/18

	$x = a^{\frac{1}{5}} + a^{- \frac{1}{5}}$ $x^{5} - 5 x^{3} + 5 x$ ${(a^{\frac{1}{5}} + a^{\frac{- 1}{5}})}^{5} - 5 {(a^{\frac{1}{5}} + a^{\frac{- 1}{5}})}^{3} + 5 (a^{\frac{1}{5}} + a^{\frac{- 1}{5}})$ $= a + 5 c_{1} a^{\frac{4}{5}} a^{\frac{- 1}{5}} + 5 c_{2} a^{\frac{3}{5}} a^{\frac{- 2}{5}} + 5 c_{3} a^{\frac{2}{5}} a^{\frac{- 3}{5}} + 5 c_{4} a^{\frac{1}{5}} a^{\frac{- 4}{5}} + 5 c_{5} a^{- 1}$ $- 5 (a^{\frac{3}{5}} + 3 c_{1} a^{\frac{2}{5}} a^{\frac{- 1}{5}} + 3 c_{2} a^{\frac{1}{5}} a^{\frac{- 2}{5}} + a^{\frac{- 3}{5)}}$ $+ 5 (a^{\frac{1}{5}} + a^{\frac{- 1}{5}})$ $= a + 5 a^{\frac{3}{5}} + 10 a^{\frac{1}{5}} + 10 a^{\frac{- 1}{5}} + 5 a^{\frac{- 3}{5}} + \frac{1}{a} +$ $- 5 a^{\frac{3}{5}} - 15 a^{\frac{1}{5}} - 15 a^{\frac{- 1}{5}} - 5 a^{\frac{- 3}{5}} + 5 a^{\frac{1}{5}} + 5^{\frac{- 1}{5}}$ $= a + \frac{1}{a} = \sqrt{3} + \sqrt{2} + \sqrt{3} - \sqrt{2} = 2 \sqrt{3} a n s w e r$
	Commented by Meritguide1234 last updated on 26/Oct/18

	$v e r y g o o d M r . T a n m o y$ $A r e y o u p r o f e s s o r ?$
	Commented by tanmay.chaudhury50@gmail.com last updated on 26/Oct/18

	$n o s i r . . i e n j o y i t . . . i h a v e i n t e r e s t i n p h y s i c s$ $a n d m a t h e m a t i c s . . .$
	Commented by Meritguide1234 last updated on 26/Oct/18

	$b a s i c a l l y i a m t e a c h e r o f s e n i o r s e c o n d a r y s c h o o l o f t a i w a n .$
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