Rationalise the complex fraction: (8 + 6i)/(6 - 2i)

The problem with this fraction is the "i" on the denominator, it is not rational. To rationalise this fraction we use the complex conjugate.

This sounds more complicated than it is. If a + ib is a complex number, then a - ib is its conjugate. The only thing that changes is that the imaginary part of the number changes its sign.

To rationalise the fraction you multiply both top and bottom of the fraction by the the complex conjugate of the denominator.

(8 + 6i)/(6 - 2i) * (6 + 2i)/(6 + 2i)

To make this less messy in text I will solve the top and bottom separately. First the top:

(8 + 6i)*(6 + 2i) = 48 + 16i + 36i + 12i^2 = 48 + 52i - 12 = 36 + 52i

Then the bottom:

(6 - 2i)*(6 + 2i) = 36 - 12i + 12i - 4i^2 = 36 + 4 = 40

Putting these together gives:

(36 + 52i)/40 

Simplified this is:

9/10 + (13/10)i