Show that `((sqrt(3))/(2)+(i)/(2))^(3)=i`

Prove that (sqrt(3)/(2) +(i)/(2))^(5) + (sqrt(3)/(2) -(i)/(2))^(5) is purely real.

show that ((sqrt(3)+i)/(2))^(6)+((i-sqrt(3))/(2))^(6)=-2

Show that ((-1+sqrt(3)i)/(2))^(n)+((-1-sqrt(3i))/(2))^(n) is equal to 2 when n is a multiple of 3 and 3 is equal to -1 when n is any other positive integer.

If z=((sqrt(3))/(2)+(i)/(2))^(5)+((sqrt(3))/(2)-(i)/(2))^(5), then

Show that ((2-i)^(2))/(3+4i)=1

Show that (i)" "{((3+2i))/((2-3i))+((3-2i))/((2+3i))} is purely real, (ii)" "{((sqrt(7)+i sqrt(3)))/((sqrt(7)-i sqrt(3)))+((sqrt(7)- i sqrt(3)))/((sqrt(7) + i sqrt(3)))} is purely real.

Prove that ((i-sqrt(3))/(-i+sqrt(3)))^(200)+((i-sqrt(3))/(i+sqrt(3)))^(200)=-1

Prove that [(i+sqrt(3))/(-i+sqrt(3))]^(100)+[(i-sqrt(3))/(i+sqrt(3))]^(100)=-1