In this guide you will find everything you need to know about surds for GCSE maths. Surds can sometimes feel like a tricky topic, but with a bit of revision and practice you’ll soon see that they’re not so bad. Let’s start with the basics…
What is a Surd?
A surd is an expression that includes a square root, cube root or any other root symbol. Surds allow us to write irrational numbers accurately.
“Okay, so what is an irrational number?” I hear you say. Well irrational numbers cannot be written as fractions. The decimal version of an irrational number does not recur or terminate. So surds are really useful because we cannot write an irrational number accurately as a decimal (we’d be writing numbers for ever). Instead we use surds to write irrational numbers precisely. Examples of surds are: √2 and 5√3
A key thing you need to be able to do with surds is to simplify them. We can do this if the number in the square root symbol has a square number as a factor.
Check out these examples to see how it works:
- √200 = √100 x √2 = 10√2
- √12 = √4 x √3 = 2√3
- √8 = √4 x √2 = 2√2
- √27 = √9 x √3 = 3√3
- √80 = √16 x √5 = 4√5
And these are the general rules you need to learn:
- √ab = √a x √b
- √a x √a = a
- √⅓ = √1/√3
Adding and subtracting surds
To add or subtract surds the numbers inside the square root symbols must be the same. If they are, then work in a similar way to collecting like terms in algebra. Check out these examples:
- √2 + √2 = 2√2
- 2√3 + √3 = 3√3
- 5√5 – 2√5 = 3√5
- 6√7 – √7 = 5√7
- 4√10 + 5√10 = 9√10
Multiplying and dividing surds
When multiplying surds with the same number in the square root we know to simplify to just the number itself. The answer will be a whole rational number. For example:
- √5 x √5 = 5
- √3 x √3 = 3
- 3√2 x √2 = 3 x √2 x √2 = 3 x 2 = 6
- (√3)2 = √3 x √3 = 3
- (√7)2 = √7 x √7 = 7
If there are whole numbers included in the question with the surds then simply multiply the numbers and the surds separately and then bring them together at the end where possible. For example:
- 3√2 x √2 = 3 x √2 x √2 = 3 x 2 = 6
- 2√5 x 4√5 = 2 x √5 x 4 x √5 = 2 x 4 x 5 = 40
When multiplying surds with different numbers in the square root, simply multiply the numbers together in a square root sign and then simplify where possible. For example:
- √5 x √10 = √50 = √25 x √2 = 5√2
- √10 x √8 = √80 = √16 x √5 = 4√5
When dividing surds we use a very similar method to that we’ve used when multiplying. Simply divide each of the component parts separately. For example:
Expanding brackets including surds
You can also be asked to multiply out brackets involving surds. When you get these questions treat them like you would an algebraic expression – use the same rules and you’ll be fine.
Take a look at the example below. Simply multiply out the bracket as you’ve learnt previously with algebra:
√3 (5 – √2) = 5 x √3 – √2 x √3 = 5√3 – √6
If you are given double brackets to multiply out, then simply use the FOIL method we’ve used before:
(√7 + 3)(√7 – 3) = √7 x √7 – 3√7 + 3√7 -9 = 7 – 9 = -2
Rationalising the denominator
Whenever we have a fraction with an irrational number on the denominator, we need to simplify by rationalising the denominator. In practice that means “getting rid of” the surd on the denominator. To do this we multiply the numerator and denominator of the fraction by the surd itself. As we’re multiplying by the surd over itself we are basically multiplying by one, so it won’t change the value of the fraction, it just changes how it looks. In doing so the denominator will be made in to a rational number. It might sound complicated, but it’s not too bad once you know what you’re doing. Take a look at the examples below to see how it all works.
Rationalising the denominator – example 1
Rationalising the denominator – example 2
Surds for GCSE Maths – become an expert
For more useful resources on surds for GCSE maths you can check out BBC bitesize. As ever, they have some excellent resources for you to use. If you would like some personalised GCSE maths lessons, or even a specific lesson on surds for GCSE maths, then you can book a trial lesson with one of our expert maths tutors by simply sending us a quick message. We’ll offer some free advice and put you in touch with the right maths tutor for you.