Question

How do you simplify `(1+3i) div (5-3i)`?

Answer 1

`-1/9+1/2i`

Explanation:

Write as `(1+3i)/(5-3i)`. Multiply by the complex conjugate of the denominator.

`(1+3i)/(5-3i)((5+3i)/(5+3i))=(5+3i+15i+9i^2)/(25cancel(-15i+15i)-9i^2)=(5+18i+9i^2)/(25-9i^2)`

Recall that `i=sqrt(-1)`, so `color(red)(i^2=-1`.

`(5+18i+9(-1))/(25-9(-1))=(-4+18i)/36=-1/9+1/2i`

Note that the answer is written in the `a+bi` form of a complex number.