If you have a binomial, in other words, a sum of two terms

It's conjugate is simply another binomial with the same numbers, but the sign separating them changes.

So the conjugate of is .

The conjugate is an extremely powerful tool because usually when you expand two binomials you get

but when you multiply conjugates, the middle terms cancel.

.

This is known as the difference of two squares.

This property comes in handy if you are dealing with surds, in particular, when you have surds in the denominator and want to rewrite it in a nicer form.

E.g.

, which is a MUCH nicer form.

A similar process is used when dividing complex numbers.

For a complex number , it's complex conjugate is .

Suppose you had .

To work out this division, you multiply top and bottom by its complex conjugate, similar to how you rationalise a denominator.

.

The other nice property of complex conjugates is the fact that when you multiply a complex number by its conjugate, you get the square of the length of the complex number itself. This makes it very easy to calculate the length of a complex number.

Let . It's conjugate is therefore

.

The length of is (by Pythagoras).

So .

Hope that helped.