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

The imaginary part of the complex number z = 3 + 4i is 4.

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

z + z* = (3 + 4i) + (3 - 4i) = 6.

z + z̅ = (3 + 4i) + (3 - 4i) = 6.

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

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

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

The conjugate of z = 4 - 3i is z* = 4 + 3i.

z = 4(cos(π/3) + i sin(π/3)) = 4(1/2 + i√3/2) = 2 + 2√3i.

The argument of z = 5 + 12i is θ = arctan(12/5).

The argument of z = 5 + 12i is given by arctan(12/5).

The imaginary part of the complex number z = 5 + 12i is 12.

z = 5(cos(π/3) + i sin(π/3)) = 5(1/2 + i√3/2) = 2.5 + 5i√3/2.

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).

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.