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...

5 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

Hamza I.

Degree: Bsc Maths and Economics (Bachelors) - LSE University

Subjects offered: Maths


“A-levels:A(>90%) in Chemistry and Maths and A in Biology. GCSEs': Achieved 12 A-A*s Currently studying BsC Maths and Economics at LSE. Hobbies include: growing vegetables, Reading, Trading platforms, Cricket. Skills/Strengths/Achievem...”

MyTutor guarantee

£18 /hr

Marco-Iulian G.

Degree: Mathematics&Computer Science (Masters) - Bristol University

Subjects offered: Maths, Further Mathematics + 2 more

Further Mathematics

“I'm in my first year at University of Bristol, studying Mathematics and Computer Science MEng. From an early age I started to participate in lots of contests and maths olympiads, and the experience I achieved along the way enriched bo...”

£36 /hr

Chris S.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Maths, Spanish+ 1 more

Further Mathematics

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

About the author

£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

How do I know which trig formula to use in the exam?

Judy bought a car for £12,000. She bought the car 4 years ago. Each year the car depreciated by 10%. How much was is the car worth now?

If 2x + y = 13 and 3x - y = 12, what are the values for x and y?

Solve the simultaneous equations: y = 4x^2 - 9x - 1 and y = 5 - 4x

View GCSE Maths tutors


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