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.
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.
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: