Let z = −2√3 + 6ia) Express z|z| in polar form.b) Determine z4
8 in polar form, in terms of its principal argument.

a) To express z in polar form, we first need to find the magnitude (r) and the argument (θ) of z.

The magnitude of z is given by |z| = √((Re(z))^2 + (Im(z))^2) = √((-2√3)^2 + (6)^2) = √(12 + 36) = √48 = 4√3.

The argument of z is given by θ = atan(Im(z)/Re(z)) = atan(-6/-2√3) = atan(√3/2) = π/3.

So, z = |z|(cos θ + i sin θ) = 4√3(cos(π/3) + i sin(π/3)).

b) To find z^4/8 in polar form, we use the properties of exponents and the fact that (r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ)).

So, z^4/8 = (4√3)^4/8(cos(4π/3) + i sin(4π/3)) = 768/8(cos(4π/3) + i sin(4π/3)) = 96(cos(4π/3) + i sin(4π/3)).

The principal argument of z^4/8 is 4π/3, which is greater than π, so we subtract 2π to get the principal argument, which is -2π/3.

So, z^4/8 = 96(cos(-2π/3) + i sin(-2π/3)) in polar form.