What is this in simplest form, (3-2i)/(6+2i)

The right answer is supposed to be (7-9i)/20

Dividing imaginary numbers can be annoying, but it's doable. :)

The thing is to treat imaginary numbers like quotients; therefore, you can't have them in the denominator.

To get rid of the quotient, multiply the top and bottom by the bottom imaginary number with the opposite sign:

(3 - 2i)(6 - 2i) = 14 - 18i (see my answer to your other question for FOILing)
(6 + 2i)(6 - 2i) uses the Difference of Squares rule, which is that (a + b)(a - b) = a^2 - b^2. So it becomes 36 - 4i^2, or 36 + 4, or 40.

This leaves you with (14 - 18i)/40, and if you divide everything by two, then you should have the correct answer.

Multiply the top and bottom by the conjugate of the denominator, which is 6-2i.

((3-2i)(6-2i)) / ((6+2i)(6-2i))

Foil the numerator and denominator:
(18-6i-12i+4i^2) / (36-4i^2)

Combine like-terms and replace (i^2) with -1:
(18-18i-4) / (36+4)
(14-18i) / 40

Divide top and bottom by two to simplify to get the final answer:
(7-9i)/20

whenever there is complex number in denominator

we rationalise the expression for simplification

so (3-2i)/(6+2i) * (6-2i)/(6-2i)

multiplying properly we get

(3-2i) * (6-2i) = 18 -4 -12i-6i =14-18i (for numerator)

and

(6+2i) * (6-2i) = 36- 4 i^2 = 40 (a^2 -b^2 =(a+b)(a-b))(denominator)

so simply answer is

(14-18i)/40 = (7-9i)/20

(3 - 2i)(6 - 2i)
---------------------
(6 + 2i)(6 - 2i)

18 - 18i - 4
---------------
36 + 4

14 - 18i
------------
40

7 - 9i
---------
20

(3-2i) * (6-2i)/(6+2i) * (6 - 2i)
18 - 6i - 12i + 4i^2 / 36 - 12i + 12i - 4i^2
14 - 18i / 40
Divide by 2:
(7-9i)/20

3-2i 6-2i
----- x------- =(14-18i)/(40)= (7-9i)/20
6+2i 6-2i