Conjugates
This word gets used in two contexts: either when dealing with binomials which are partly rational and partly irrational, such as
and other times when dealing with binomials which are partly rational and partly imaginary, such as
(If you haven't studied imaginary numbers so far, then don't think about the second kind yet.)
When the first type of binomial occurs in the denominator of a fractions, conjugates are used to rationalize the denominator .
The conjugate of is , and the conjugate of is .
Example 1:
Simplify.
Multiply both the numerator and denominator by the conjugate of the denominator.
The denominator is now a difference of squares.
Use the power of a product property in the denominator.
Conjugates of the second type are used to divide complex numbers.
Example 2:
Simplify.
Multiply both the numerator and denominator by the conjugate of the denominator.