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Let z = ((sqrt(3))/(2) + (i)/(2))^(5)+((...

Let `z = ((sqrt(3))/(2) + (i)/(2))^(5)+((sqrt(3))/(2)-(i)/(2))^(5)`. If R(z) and I(z), respectively, denote the real and imaginary parts of z, then

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We have `((sqrt(3))/(2) +(i)/(5))^(5) +((sqrt(3))/(2) -(i)/(2))^(5)`
`= (sqrt(3)/(2) + (i)/(2))^(5) + (bar(sqrt(3)/(2) + (i)/(2)))^(5)`
`=(sqrt(3)/(2) + (i)/(5))^(5) + ((bar(sqrt(3)/(2) + (i)/(2)))^(5)) (because (barz)^(n) = bar((z^(n))))`
`= z + barz`, where `z = (sqrt(3)/(2) + (i)/(2))^(5)`
`= 2Re(z)`
Hence, given complex number is real.
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