# Can someone help me with this algebra question (imaginary numbers)?

The answer to this is a), all i dont understand is how to solve for the Theta (θ) ? can someone please explain that to me, Thank You!

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ),

where, r = | z | = √(a2 + b2).

if a > 0, then θ = tan- 1(b / a), and

if a < 0, then θ = tan- 1(b / a) + π or θ = tan- 1(b / a) + 180o.

The complex number is z = 4(1 + i√3) = 4 + 4√3i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here, a = 4 > 0 and b = 4√3.

So, first find the absolute value of r .

r = | z | = √(a 2 + b 2)

= √[ 42 + (4√3)2 ]

= √[16 + 48]

= √64

= 8.

Now find the argument θ.

Since a = 4 > 0, use the formula θ = tan- 1(b / a).

θ = tan- 1[ (4√3)/4 ]

θ = tan- 1(√3)

θ = 600

θ = 60 * π/180 radians

The polar form of z is about  8[ cos (π/3) + i sin (π/3) ].

By euler's equation, we obtain the polar form as : z = re iθ.

z = 8ei(π/3) .

w = z1/3 = [ 8ei(π/3) ]1/3 = 2e(π/9)i .

Therefore, w = 2e(π/9)i .

Option d is the correct choice.