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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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Given that
`z=((sqrt(3))/(2)+i(1)/(2))^(5)+((sqrt(3))/(2)-i(1)/(2))^(5)`
`=[cos((pi)/(6)+isin((pi)/(6))]^(5)+[cos((pi)/(6))-isin((pi)/(6))]^(5)`
`=cos^(5pi)/(6)+isin(5pi)/(6)+cos(5pi)/(6)-isin(5pi)/(6)`
Hence, Im(z)=0.
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