This blog will be a guide to rationalising the denominator of a fraction. It will cover all types of these questions that can come up in GCSE mathematics. Please note that this topic is only examinable on the higher tier GCSE, so if you are studying foundation this blog will not be applicable to you.
What is a surd?
Simply put, a surd is a square root of a non-square number. A square number is a number that is another integer (whole number) squared.
For example, 25 is a square number because it is 52 and 16 is a square number because it is 42. But 18 is not a square number because there is no integer that squares to make 18. Another way of checking a square number is to square root it. When you square root 18, you do not get a whole number answer (you get 4.24264068711928). So this means 18 is not a square number.
So, an example of a surd would be the square root of 18. But the square root of 16 would not be a surd because 16 is a square number and so its root can just be written as 4, which is an integer.
What does rationalising the denominator actually mean?
First let’s define a couple more terms.
The denominator of a fraction is the number on the bottom of the fraction. For example, 4 is the denominator of the fraction 3/4.
An irrational number is a number that cannot be written as a fraction. You don’t really need to know this at GCSE level, but what is helpful is to know that certain numbers are irrational. The most common examples of irrational numbers in GCSE maths are surds and pi.
So, when we are asked to rationalise the denominator, we need to change the fraction so that the bottom number is not irrational. In layman’s terms, it means “get rid of the irrational number from the bottom”.
Below are some examples of fractions which have irrational denominators:
At GCSE level, you will only need to do this when the denominator is a surd – and these will be the examples we look at.
Example 1
Rationalise the denominator of
We need to change the denominator of the fraction, without changing the value of the
This is very easy to do. All we need to do is multiply the fraction by the surd over itself.
Like this:
Why do we do this? Well, as you may already know, any number over itself is always 1. And multiplying any number by 1 does not change the value of the number. So we can do this and it will not change the value of the fraction.
Multiplying the numerators:
Multiplying the denominators:
The square root of 9 is 3, so this means the denominator is now 3.
So the fraction is now:
The fraction has a rational denominator, so we have done it. We have rationalised the denominator.
Example 2
Rationalise the denominator of the fraction
We would do the same here. You can ignore the 3 in the second fraction and just multiply by the surd over itself.
Multiplying the numerators:
Multiplying the denominators:
Again, the denominator has become an integer. So we have rationalised the denominator. The final answer is below.
Example 3
Rationalise the denominator of the fraction
Again, we do the same.
When multiplying the numerators, we will need to multiply out a bracket, like this:
Multiplying the denominators:
So the final answer is:
Example 4
Rationalise the denominator of the fraction
This time, the denominator consists of a surd and an integer. This is a bit trickier and will require more thought. If we try the same tactic we used with examples 1 to 3, it’s not going to work unfortunately. It won’t actually eliminate all the surds from the denominator. Take a look:
We still have a surd on the bottom of the fraction. So that didn’t work.
Instead, we need to do this:
This may seem like a strange choice, but look what happens with the denominator…
You noticed that the surd terms have cancelled out, leaving the denominator as just 7. So this has done it. Multiplying the numerators, the final rationalised denominator is:
You may recognise this little trick. We have used the difference of two squares. You can read more about this in a previous blog.
So, when the denominator contains a surd and a number, the trick is to multiply by the denominator, but with the sign switched. So either from + to – or from – to +.
I hope you have found that useful and that you now know what rationalising the denominator is and how to do it.
For more practice of this skill, you can try our surds worksheet.
If you are in Bristol and need more help with this topic or any other aspect of mathematics, book in a free taster session.
