Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).
The modulus |z| = r, as |re^(iθ)| = r.
The equation of the circle with radius 10 is x^2 + y^2 = 10^2 = 100.
Using the binomial expansion, z^3 = (x + yi)^3 = x^3 - 3xy^2 + (3x^2y - y^3)i.
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i + 3 = 5 - i.
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i - 3i^2 = 2 - i + 3 = 5 - i.
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - 1 = 3 + i.
z1 + z2 = (2 + 3i) + (4 - 5i) = 6 - 2i.
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = -3π/4.
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - i^2 = 2 + 1 + i = 3 + i.
The real part of z = 4 - 3i is 4.
(1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2.
The argument of z = -1 + 0i is arg(z) = π.
The conjugate of z = 2 + 5i is z̅ = 2 - 5i.
The conjugate of z = 5 - 2i is 5 + 2i.
The conjugate of z = 5 - 6i is 5 + 6i.
The modulus |z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.
(1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2.
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 + 1 = 2.
The real part of 5 - 7i is 5.
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
The real part is Re(z) = 4 * cos(π/3) = 4 * 1/2 = 2.
The real part of z = 5 - 2i is 5.
The real part of z = 5 - 4i is 5.
The real part of z = 5 - 6i is 5.
