__Definition__

**Square root of
the complex number z = a + bi, is called the complex number w = x + yi
for which it is valid that or .**

__Example__

Find the square root of z=5-12i.

__Solution__

The square root of
z is the complex number x + yi for which

We have:

So, or
2xy = 12 (1)
or xy = 6 (2)

and

which
is impossible or

If x=3 the equation (2) gives y = 2

If x=-3 the equation (2) gives y = -2

Therefore the number z = 5 - 12i has two
square roots: 3 + 2i, -3 - 2i

If z = a + bi a complex number and M(a,b)
the corresponding element on the Complex plane.

If
and then and .

Thus
**Trigonometric form of z**

The angle is
called argument of z and is denoted arg(z).

Now, if , then easily and not using much trigonometry, it can be proved that:

Also the following can be proved:

__De Moivre's
Theorem__

**If n integer and a
complex number, then **

For example:

.
** **