Show that`((19 + 9i)/(5 - 3i))^(15) - ((8 + i)/(1 + 2i))^(15)` is purely imaginary

Show that ((19- 7i)/(9 + i))^(12) + ((20 - 5i)/(7 - 6i))^(12) is real.

Show that ( 2 + i sqrt(3))^(10) - (2 - i sqrt(3))^(10) is purely imaginary.

Prove that : (1 + i)^(4n) and (1 + i)^(4n + 2) are real and purely imaginary respectively.

Let A={theta in (-pi /2,pi):(3+2i sin theta )/(1-2 sin theta ) is purely imaginary } Then the sum of the elements in A is

Show that ((sqrt(3))/(2) + (i)/(2))^(5) + ((sqrt(3))/(2) - (i)/(2))^(5) = -sqrt(3)

Show that : (i) {i^(19) + (1/i)^(25)}^(2) = -4 " "(ii) {i^(17) - (1/i)^(34)}^(2) = 2i (iii) {i^(18) + (1/i)^(24)}^(3) = 0 " " (iv) i^n + i^(n+1) + i^(n +2) + i^(n + 3) = 0 , for all n in N .

The conjugate of i^(13) + i^(14) + i^(15) + i^(16) is …………

Reduce (4+3i)/(5-4i) in x+iy form, find its real and imaginary parts.