For the first root, we need to find `sqrt(-5+12j`.

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

We express −5 + 12j in polar form:

`r=sqrt((-5)^2+12^2=13`

For the angle:

`alpha=tan^-1(y/x)`

`=tan^-1(12/5)~~67.38^text(o)`

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=(r^n∠ nθ)`,

we have:

`(-5+12j)^(1"/"2)`

`=13^(1"/"2)/_(1/2xx112.62^@)`

`=3.61/_56.31^@`

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 `360^text(o)/n` 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`.