Question

If `z=(sqrt(3)/2+i/2)^5+(sqrt(3)/2-i/2)^5` , then prove that `Im(z)=0`

Answer 1

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.