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The first thing you need to do is rationalize the denominator. To rationalize a denominator containing an irreduceable binomial, multiply the entire fraction by 1 in the form of the conjugate of the denominator divided by itself.
If you have a complex number of the form
, the conjugate is
The product of the denominators will then be a single rational number because the product of
(assuming rational coefficients a and b in the complex number).
Now use FOIL to multiply numerator times numerator, and use the reverse of the difference of two squares factorization to multiply denominator times denominator. Then collect like terms. If your result has a common integer factor in both numerator and denominator, take that out.