For the first root, we need to find .

This is the same as (-5 + 12j)^1/2.

We express -5 + 12j in polar form:

r =

For the angle:

The complex number -5 + 12j is in the second quadrant, so

θ = 180° - 67.38 = 112.62°

So -5 + 12j = 13 ∠ 112.62°

Using DeMoivre's Theorem:

(r ∠ θ)ⁿ = (rⁿ∠ nθ),

we have:

This is the first square root. In rectangular form,

x = 3.61 cos56.31° = 2

y = 3.61 sin56.31° = 3

So the first root is 2 + 3j.

CHECK: (2 + 3j)² = 4 + 12j - 9 = -5 + 12j Checks OK.

To obtain the other square root, we apply the fact that if we need to find n roots they will be apart.

In this case, n = 2, so our roots are 180°apart.

Adding 180° to our first root, we have:

x = 3.61 cos(56.31° + 180°) = 3.61 cos(236.31°) = -2

y = 3.61 sin(56.31° + 180°) = 3.61 sin(236.31°) = -3

So our second root is -2 - 3j.

So the two square roots of -5 - 12j are 2 + 3j and -2 - 3j.