Will you explain the concept of "i" in algebra?

Question

Problem: 4+2i/4-2i

Answer 1

i = sqrt (-1)
so i^2 = sqrt (-1)(sqrt(-1) = -1

(4+2i)/(4-2i)

to get rid of i in the denominator multiply top and bottom by the "comlex conjugate of (4-2i) which is (4+2i)

Rule: conjugate of (a-bi) is (a+bi)

(4+2i)(4+2i)
-------------
(4-2i)(4+2i)

16 + 16 i +4 i^2
=-------------------
(16 - 2 i^2)

but we all know that i^2 = -1 so

12 + 16 i
----------
18

or

6 + 8 i
--------
9