|z|^2 = (1)^2 + (√3)^2 = 1 + 3 = 4.

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

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

|z|^2 = 1^2 + (√3)^2 = 1 + 3 = 4.

The argument is given by tan^(-1)(√3/1) = π/3.

The modulus of z = 1 + √3i is |z| = √(1² + (√3)²) = √(1 + 3) = √4 = 2. The square of the modulus is 2² = 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.

|z| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.

z/z* = (2 + 2i)/(2 - 2i) = (2 + 2i)(2 + 2i)/(4 + 4) = (4 + 8i - 4)/(8) = i.

z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = -8 + 4i.

|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.

z² = (2 + 2i)² = 4 + 8i - 4 = 8i.

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

z^2 = (2 + 2i)^2 = 4 + 8i - 4 = 8i.

z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = 8.

|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.

The conjugate of z is 2 - 3i.

The argument of z = 2 + 3i is θ = arctan(3/2).

The conjugate of z = 2 - 3i is 2 + 3i.

z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.

|z| = 2, as |z| = r where z = r(cos(θ) + i sin(θ)).

z = 2(1/2 + i√3/2) = 1 + √3i.

Using Euler's formula, z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.

z = 2(cos(π/4) + i sin(π/4)) = 2(√2/2 + i√2/2) = √2 + √2i.

|z| = 2, as |r(cosθ + isinθ)| = r.

z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.

z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3.

z^2 = (2e^(iπ/4))^2 = 4e^(iπ/2).

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