# Surds

## Simplifying Surds

**What is a surd?**

A surd is a number that is irrational. It cannot be written exactly. For example **√3**= 1.732051080……. It continues indefinitely and cannot be written as a fraction.

**Simplifying Surds**

When working with surds there are certain rules we need to learn to help us manipulate and simplify them:

A key thing to remember with surds is that if you can get to a square number, then you can simplify the surd!

Always aim to make square numbers under surds. This usually involves breaking a surd down into its square factors.

For example to simplify √60

Write down the factors of 60 and look for square factors

1 x 60 , 2 x 30, 3x 20, __4 x 15, __5 x 12, 6 x 10

4 is a square number so now we have a square factor!

**√60 __= __√4** x **√15 __= 2**√15__

15 does not have any square factors (1 x 15, 3 x 5) therefore we cannot simplify the surd anymore.

= 2**√15** is our answer.

## Rationalising Denominators

A key question with surds often involves rationalising a denominator.

If you remember that a surd is a square root then **√** a x **√** a = a

Given this sometimes you will be required to rationalise the denominator of a fraction, if it is a surd. Eg.

By making the denominator 2 you have rationalised it!

The same logic applies for more complicated examples: