Conjugates

This word gets used in two contexts: either when dealing with binomials which are partly rational and partly irrational, such as

$3+5\sqrt{2}$

and other times when dealing with binomials which are partly rational and partly imaginary, such as

$4-2i$

(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 $a+\sqrt{b}$ is $a-\sqrt{b}$ , and the conjugate of $a+bi$ is $a-bi$ .

Multiply both the numerator and denominator by the conjugate of the denominator.

$\frac{1}{4-3\sqrt{7}}=\frac{1}{4-3\sqrt{7}}\cdot \frac{4+3\sqrt{7}}{4+3\sqrt{7}}$

The denominator is now a difference of squares.

$=\frac{4+3\sqrt{7}}{4^2-{\left(3\sqrt{7}\right)}^2}$

Use the power of a product property in the denominator.

$=\frac{4+3\sqrt{7}}{4^2-\left(9\cdot 7\right)}$

$=\frac{4+3\sqrt{7}}{16-63}$

$=\frac{-4-3\sqrt{7}}{47}$

Conjugates of the second type are used to divide complex numbers.

Multiply both the numerator and denominator by the conjugate of the denominator.

$=\frac{-2+i}{5-2i}\cdot \frac{5+2i}{5+2i}$

$\begin{array}{l}=\frac{-12+i}{29}\\ =-\frac{12}{29}+\frac{1}{29}i\end{array}$