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

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

We express −5 + 12*j* 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 + 12*j* 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 + 3*j*.

**CHECK:** (2 + 3*j*)^{2} = 4 + 12*j* - 9
= -5 + 12*j * [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`.