# Conjugate

The conjugate is where we **change the sign in the middle** of two terms like this:

**two terms**, called "binomials":

example of a binomial |

Here are some more examples:

Expression | Its Conjugate | |
---|---|---|

x^{2} − 3 |
⇔ | x^{2} + 3 |

a + b | ⇔ | a − b |

a − b^{3} |
⇔ | a + b^{3} |

## Examples of Use

The conjugate can be very useful because ...

... when we multiply something by its conjugate we get **squares** like this:

^{2}− b

^{2}

### How does that help?

It can help us move a square root from the bottom of a fraction (the *denominator*) to the top, or vice versa. Read Rationalizing the Denominator to find out more:

**Example:** Move the square root of 2 to the top:

\frac{1}{3−√2}

We can * multiply both top and bottom by 3+√2 (the conjugate of 3−√2)*, which won't change the value of the fraction:

\frac{1}{3−√2} × \frac{3+√2}{3+√2} = \frac{3+√2}{3^{2}−(√2)^{2}} = \frac{3+√2}{7}

(The denominator becomes **(a+b)(a−b) = a ^{2} − b^{2}** which simplifies to 9−2=7)

*Use a calculator to work out the value before and after ... is it the same?*

So try to remember this little trick, it may help you solve an equation one day!