The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.

The conjugate of z = 5 - 6i is z̅ = 5 + 6i.

The real part of z = 2 + 3i is 2.

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

The real part of z is 3.

The real part of z = 4 + 3i is 4.

The real part is 4 * cos(π/3) = 4 * 1/2 = 2.

The real part of z = 5 - 2i is 5.

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

(1 + i)^4 = (√2 e^(iπ/4))^4 = 4 e^(iπ) = 4(-1) = -4.

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

Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.

Taking square root, z = ±√(-16) = ±4i.

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

The conjugate of z = 1 + i is z̅ = 1 - i.

z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.

z^4 = (1 + i)^4 = (2e^(iπ/4))^4 = 16e^(iπ) = -16.

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

z^3 = (1 + i)^3 = 1 + 3i - 3 - i = -2 + 2i.

z^4 = (1 + i)^4 = 4i.

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

The argument θ = tan^(-1)(√3/1) = π/3.

|z| = √(1^2 + (√3)^2) = √(1 + 3) = √4 = 2.

|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.

|z| = √(1^2 + (√3)^2) = √(1 + 3) = √4 = 2.

|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.

The argument θ = tan^(-1)(√3/1) = π/3.

|z|^2 = 1^2 + (√3)^2 = 1 + 3 = 4.

The argument is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.

The conjugate of z = 2 + 2i is z̅ = 2 - 2i.