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.

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

The conjugate of z is 2 - 3i.

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

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.

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.

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

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

The modulus |z| = √(a^2 + b^2) = 10 implies a^2 + b^2 = 100.

The conjugate of z = a + bi is z̅ = a - bi.

The conjugate of z = a + bi is z̅ = a - bi.

The conjugate of z = a + bi is z̅ = a - bi.

Using De Moivre's theorem, z^4 = (cos(θ) + i sin(θ))^4 = cos(4θ) + i sin(4θ).

z^8 = (e^(iπ/4))^8 = e^(i2π) = 1.

The modulus |z| = r in the polar form z = re^(iθ).

r = |z| = √(1^2 + 1^2) = √2.

r = |z| = √(3^2 + 4^2) = √25 = 5.

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

r = |z| = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2.

Using the property of exponents, z^2 = (re^(iθ))^2 = r^2e^(i2θ).

Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).