The real part is Re(z) = 5cos(π/3) = 5 * 1/2 = 5/2.

The sum is (2 + 4) + (3 - 5)i = 6 - 2i.

(1 + i)^2 = 1^2 + 2*1*i + i^2 = 1 + 2i - 1 = 2i.

i^4 = (i^2)^2 = (-1)^2 = 1.

The solutions to z^2 = -1 are z = i and z = -i.

|z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.

|z| = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.

The modulus |z| = 4, as it is the coefficient in the polar form.

|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25.

A purely imaginary number has no real part, so 0 + 5i is purely imaginary.

The roots are z = ±i.