Question

How do you simplify `(18+3i)/(18-3i)`?

Answer 1

`\frac{35+12i}{37}`

Explanation:

First of all, you can simplify everything by a factor `3`:

`\frac{18+3i}{18-3i}=\frac{6+i}{6-i}`

Then, you can do something similar to razionalization with roots: multiply and divide by `6+i`:

`\frac{6+i}{6-i} \frac{6+i}{6+i} = \frac{(6+i)^2}{(6+i)(6-i)}`

Now use the fact that `(a+b)(a-b)=a^2-b^2`:

`\frac{(6+i)^2}{6^2 - i^2}`

Now remember that `i^2=-1`:

# \frac{(6+i)^2}{6^2 +1} =\frac{36+12i-1}{37}=\frac{35+12i}{37}#