# Conjugate

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

We only use it in expressions with **two terms** (called "binomials")

### Other 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:

### 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}

Let's * multiply both top and bottom by the conjugate* (this will not 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 ^{2} − b^{2}** which simplifies to 9−2=7)

Get your calculator and work out the value before and after ... is it the same?

There is another example on the page Evaluating Limits (advanced topic) where I move a square root from the top to the bottom.

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