if x^2 −x+1 = 0 and α and β are thd roots of this equation then evaluate ((α^(100) +β^(100) )/(α^(100) −β^(100) ))


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Question Number 191611 by vishal1234 last updated on 27/Apr/23

$i f x^{2} - x + 1 = 0 a n d α a n d β a r e t h d r o o t s$ $o f t h i s e q u a t i o n t h e n e v a l u a t e \frac{α^{100} + β^{100}}{α^{100} - β^{100}}$
	Answered by mehdee42 last updated on 27/Apr/23

	$x = \frac{1 \pm \sqrt{3} i}{2} \Rightarrow α = \frac{1 + \sqrt{3} i}{2} = e^{\frac{π}{3} i} & β = \frac{1 - \sqrt{3} i}{2} = e^{- \frac{π}{3} i}$ $α^{100} + β^{100} = e^{\frac{100 π}{3} i} + e^{- \frac{100 π}{3} i} = 2 c o s \frac{100 π}{3} = - 2 c o s \frac{π}{3} = - 1$ $α^{100} - β^{100} = e^{\frac{100 π}{3} i} + e^{- \frac{100 π}{3} i} = 2 s i n \frac{100 π}{3} = - 2 i s i n \frac{π}{3} = - \sqrt{3} i$ $\Rightarrow a n s w e r = \frac{1}{\sqrt{3} i} = - \frac{\sqrt{3}}{3} i$
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