How do I rationalise the denominator of a fraction?

I am going to assume you know what an irrational number is, given that that should have been taught right at the beginning the topic of surds. When I talk about a “rational” number, here I mean “whole number” or “integer”, although a rational number is a number that is not irrational.

There are two types of problem that you could come across when being asked to rationalise the denominator of a fraction:

The first type is when you are asked to rationalise a fraction which has its whole denominator under a square root. For example 1/sqrt(2) or 5/sqrt(8) etc. To rationalise the denominator here, we somehow have to square the denominator to get a whole number. However, we can’t change the value of the fraction! A great way of doing this is therefore by multiplying the fraction with denominator/denominator as you’ll notice that this is just equal to 1: Let’s say we had the fraction n/d (n for numerator, and d for denominator), if we multiply this by d/d, we get nd/d2 (which is still equal to our original fraction). If ‘d’ was irrational, e.g. d=sqrt(2), then d2 would be rational, e.g. (sqrt(2))2=2, and therefore we have successfully rationalised the denominator! For example, we could multiply 1/sqrt(2) by sqrt(2)/sqrt(2) to get sqrt(2)/( sqrt(2))2 = sqrt(2)/2, or 5/sqrt(8) by sqrt(8)/ sqrt(8) to get 5sqrt(8)/(sqrt(8))2 = 5sqrt(8)/8.

The second type is a little trickier as it relies on a concept we call “the difference of two squares”. First, let’s consider (x+y)(x-y) = x2-y2; this result is known as “the difference of two squares”, and it is incredibly useful because if either ‘x’ or ‘y’ were irrational, then we now know that (x+y)(x-y) is rational. For example, if x is irrational (e.g. x=sqrt(2)) and y is rational (e.g. y=3), then (x+y)(x-y)= x2-y2 =(sqrt(2))2-32=2-9=-7, or if x is rational (e.g. x=4) and y is irrational (e.g. y=sqrt(5)), then (x+y)(x-y)= x2-y2=42-(sqrt(5))2=15-5=11. Now let’s return to the second type of problem: you could be asked to rationalise the denominator of a fraction whose denominator comprises of a rational number added to a whole number, for example 1/(1+sqrt(2)) or 3/(sqrt(3)-5) etc. This is definitely more tricky than the first type of problem, but here is where we use what we’ve just learned about “the difference of two squares” to make life a lot easier for us: The denominators of these fractions look a lot like the examples I gave for (x+y) or (x-y) (so 1+sqrt(2) is just (x+y) where x=1 and y=sqrt(2) etc.) so, we probably want to find a way to multiply the denominator of the fraction by the ‘opposite form’ in order to rationalise it. For example, multiplying (1+sqrt(2)) by (1-sqrt(2)) gives us the difference of two squares: 12-(sqrt(2))2=1-2=-1, and multiplying (sqrt(3)-5) by (sqrt(3)+5) = (sqrt(3))2-52=3-25=-22. Now we have a way of turning the denominator into a whole number, we are back to thinking about how we can do this without changing the value of the fraction. Using our trick from the first type, you can see that, if we were given a fraction a/(b+sqrt(c)), then multiplying it by (b-sqrt(c))/ (b-sqrt(c)) wouldn’t change the value ( as (b-sqrt(c))/ (b-sqrt(c))=1) but gives us b2-(sqrt(c))2 on the bottom, which is b2-c, which is a whole number! Going back to my earlier examples, 1/(1+sqrt(2)) x(1-sqrt(2))/ (1-sqrt(2)) = (1-sqrt(2))/-1 = sqrt(2)-1, and 3/(sqrt(3)-5) x(sqrt(3)+5)/ (sqrt(3)+5)=3(sqrt(3)+5)/-22 = (3sqrt(3)+15)/-22 =(-3sqrt(3)-15)/22.

Andrew D. A Level Maths tutor, GCSE Maths tutor, A Level Further Math...

9 months ago

Answered by Andrew, a GCSE Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£18 /hr

Dan W.

Degree: Economics and Accounting (Bachelors) - Bristol University

Subjects offered:Maths, Economics


“I achieved top grades whilst juggling cricket at a high level. I’ve tutored for Young Einstein Tuition & been a Peer Mentor to those facing personal issues”

£18 /hr

Bukky O.

Degree: Maths with Finance (Bachelors) - Exeter University

Subjects offered:Maths, Chemistry


“Hi, I'm Bukky (pronounced book-ee) and my aim is to not only help your child excel in Maths but also hopefully enjoy it too”

£22 /hr

Haider M.

Degree: Medicine (Bachelors) - Exeter University

Subjects offered:Maths, Physics+ 7 more

Extended Project Qualification
.BMAT (BioMedical Admissions)
-Personal Statements-
-Medical School Preparation-

“A bit about me I am currently in my 2nd year of medical school at the University of Exeter. I really enjoy learning about science, especially the science of health disease, and would love nothing more than to pass down this knowledge ...”

About the author

Andrew D. A Level Maths tutor, GCSE Maths tutor, A Level Further Math...
£18 /hr

Andrew D.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:Maths, Further Mathematics

Further Mathematics

“About me Hi, I'm Andrew. I'm currently going into my second year studying maths at the University of Warwick. I find maths incredibly fascinating, as my passion stems from the idea thatmaths is everywhere and is fundamental to underst...”

You may also like...

Posts by Andrew

How do I find the limit as x-->infinity of (4x^2+5)/(x^2-6)?

How do I rationalise the denominator of a fraction?

How do I solve an integration by substitution problem?

How do I use proof by induction?

Other GCSE Maths questions

What is the difference between distance and displacement?

What are the possible value(s) of x for the following: x^2 + 3x - 54 = 0

How do you work out the old price of an item having been given the new price after a specified percentage change?

How do I solve the inequality 7x+2 > 2x-3?

View GCSE Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss