MYTUTOR SUBJECT ANSWERS

438 views

How do you 'rationalise the denominator'?

There are two approaches to rationalising a denominator depending on how the denominator appears.

First we deal with the simpler case where the surd term is the only term in the denominator.

For example: 6/√3, 4/(3√7) or (5+√2)/√3

The trick of 'multiplying by one' can be used on each of these fractions to give an alternate expression that does not contain a surd in the denominator.

Let's consider the fraction 6/√3

If we multiply this fraction by √3/√3 we do not change the value of the fraction because √3/√3=1

Let's go ahead and do the multiplication:

The numerator is 6* √3 which is most simply expressed as 6√3

The denominator is √3*√3 which by the definition of the square root function is equal to 3

So putting the numerator and denominator together leaves us with the fraction (6√3)/3

Some basic cancellation reduces this to the simpler form of 2√3

So 6/√3=2√3

Next comes the more complicated case where the surd term is not the only term in the denominator.

For example: 1/(1-√3), (1+2)/(3+2) or 6/(25+3)

The trick of 'multiplying by one' can again be used but it is less clear what 'one' to use.

To find our 'one' we must recall the difference of two squares formula: a2-b2=(a+b)(a-b)

Let's consider 1/(1-√3)

From the difference of two squares formula we can see that if we multiply the denominator by 1+√3 then we are left with 12-(√3)2=1-3=-2

From this we can infer that it is necessary to multiply the fraction by (1+√3)/(1+√3)

Let's go ahead and do the multiplication:

The numerator is 1*(1+√3) which is equal to 1+√3

The denominator we have already calculated is -2

So putting the numerator and the denominator together leaves us with (1+√3)/-2

So 1/(1-√3)=(1+√3)/-2

A useful reason for rationalising the denominator is that it helps when thinking about what value a fraction really represents. For example, when considering the fraction 1/√2=√2/2, it is hard to imagine 1 divided into √2 pieces as √2 is an irrational number. It makes more sense however to imagine √2 divided into 2 pieces as that is just a simple halving and it doesn't matter too much that √2 is irrational.

Jacob G. A Level Maths tutor, A Level Further Mathematics  tutor, GCS...

1 year ago

Answered by Jacob, an A Level Maths tutor with MyTutor


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

183 SUBJECT SPECIALISTS

£20 /hr

Angus S.

Degree: Cognitive Science (BSc Hons) (Bachelors) - Edinburgh University

Subjects offered: Maths

Maths

“Hello! My name is Angus and I'm 19 years old. Currently, I'm a first year Informatics student at the University of Edinburgh. I spent part of a year after school travelling and teaching English as a foreign language, although Maths wa...”

MyTutor guarantee

£20 /hr

Peter H.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.STEP.

“About me: I am a Mathematics student at Durham University, and I am currently in my 3rd year. I would very much like to give assistance and help students of all abilities to reach the goals they aspire to in Maths and Physics, whether...”

MyTutor guarantee

£20 /hr

Louise S.

Degree: Mathematics with Education & languages (Bachelors) - Imperial College London University

Subjects offered: Maths, Spanish+ 4 more

Maths
Spanish
Physics
Further Mathematics
Chemistry
.MAT.

“I am a 2nd year maths student at imperial. My aim is for students to understand what they are learning, not just to learn it by rote. I think learning should be fun , and I will try and make them so.”

MyTutor guarantee

About the author

Jacob G.

Currently unavailable: for regular students

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered: Maths, History+ 2 more

Maths
History
Further Mathematics
English

“About Me: Hi, I'm Jacob and I'm a second-year student of Mathematics at the University of Warwick. My main subject isMaths and I averaged 98.5% over my Maths and Further Maths A-level exams. I also maintain an interest inEnglish and Hi...”

MyTutor guarantee

You may also like...

Posts by Jacob

How do you 'rationalise the denominator'?

How do you find the square roots of a complex number?

Other A Level Maths questions

Differentiate the equation y = x^2 + 3x + 1 with respect to x.

Find minimum and maximum of x^2+1 if they exist

What is a Probability Mass Function (PMF)?

How to factorise any quadratic expression

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok