What is the value of `((i+sqrt3)/(-i+sqrt3))^(200)+((i-sqrt3)/(i+sqrt3))^(200)+1`?

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

B

Explanation

Consider, `(i+sqrt3)/(-i+sqrt3)=((i+sqrt3)^(2))/((sqrt3-i)(sqrt3+i))` (By Rationalizing)
`=(i^(2)+3+2sqrt3i)/(3+1)" "("using"(a-b)(a+b)=a^(2)-b^(2))`
`=(-1+3+sqrt3i)/(4)=(1+sqrt3i)/(2)`
`=(-(-1-sqrt3i))/(2)=-omega^(2)" "(becausei^(2)=-1)`
and consider
`(i-sqrt3)/(i+sqrt3)=((i-sqrt3)^(2))/(i^(2)-(sqrt3)^(2))` (By Rotationalizing)
`=(i^(2)+3-2isqrt3)/(-4)=(2-2isqrt3)/(-4)(becausei^(2)=-1)`
`=(-1+isqrt3)/(2)=omega`
`therefore((i+sqrt3)/(-i+sqrt3))^(200)+((i-sqrt3)/(1+sqrt3))^(200)+1`
`=(-omega^(2))^(200)+omega^(200)+1`
`=omega^(400)+omega^(200)+1=omega^(3xx133+1)+omega^(3xx66+2)+1`
`=(omega^(3))^(133)omega+(omega^(3))^(66)omega^(2)+1`
`=omega+omega^(2)+1" "(becauseomega^(3)=1)=0`