# 1.5.1 Surds - Basics

#### What is a surd?

• A surd is the square root of a non-square integer

#### What can we do with surds?

1. Multiplying surds – you can multiply numbers under square roots
eg. √3 × √5 = √3×5 = √15

2. Dividing surds – you can divide numbers under square roots
eg. √21 ÷ √7= √21 ÷ 7 = √3

3. Factorising surds – you can factorise numbers under square roots
eg. √35 = √5 × 7 = √5 × √7

4. Simplifying surds – separate out a square factor and square root it!
eg. √48 = √16 × 3 = √16 × √3 = 4 × √3 = 4√3

5. Adding or subtracting surds is very like adding or subtracting letters in algebra – you can only add or subtract multiples of “like” surds
eg. 3√5 + 8√5 = 11√5 or 7√3 – 4√3 = 3√3
Be very careful here! You can not add or subtract numbers under square roots. Think about √9 + √4= 3 + 2 = 5. It is not equal to √9 + 4= √13 = 3.60555…

6. All other algebraic rules apply – surds can be treated like letters (as in 5. above) and like numbers (as in 1. and 2. above)

### Worked Example

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