MYTUTOR SUBJECT ANSWERS

855 views

How do I rationalise the denominator of a fraction which consists of surds?

[Recall: the numerator of a fraction is the top number; the denominator refers to the bottom number. 

A surd is an irrational number, e.g. √3, √5, etc.]

Given the following fraction:

(a+√b)/(c-√b), where a,b and c are non-negative integers and b is not a square number.

We can see that the denominator (c-√b) is irrational. To rationalise the denominator we take the following steps:

1. Multiply BOTH the numerator and the denominator of our fraction by (c+√b) in order to eliminate the irrational surd in the denominator. 

Note: we perform this multiplication to both the numerator and denominator in order to preserve the value of the original fraction .

2. We now have for our numerator: (a+√b)(c+√b), and for our denominator: (c-√b)(c+√b). 

Expand these brackets, thus we obtain the following fraction:

(ac+(a+c)√b+b) / (c- b)

Clearly we have succeeded in rationalising our denominator (whilst still maintaining the value of our original fraction) since (c2-b) is clearly a rational number, as required.

Example:

Write (5+7√3)/(5-3) in the form a+b3, where a and b are rational.

Soln: We carry out the steps stated above;

Multiply numerator and denominator by (5+√3), in doing so eliminating the irrational surd from our denominator. We thus obtain:

{(5+7√3)(5+√3)} / {(5-√3)(5+√3)}  (expand brackets)

=(46+12√3) / (25+5√3-5√3 - 3)

=(46+12√3) / (22)

=23/11+(6/11)√3.

Clearly, from our orginal hypothesis, a=23/11, b=6/11 are both rational numbers, thus we are done.

Liam D. GCSE Maths tutor, A Level Maths tutor

1 year ago

Answered by Liam, an A Level Maths tutor with MyTutor

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

182 SUBJECT SPECIALISTS

£20 /hr

Amilah C.

Degree: Physics with a year abroad (Masters) - Edinburgh University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Chemistry

“Hi there, welcome to me page! About Me: I am currently studying Physics at the University of Edinburgh. From a young age, science has always been a passion of mine, and hopefully I can convey this in my tutorials. Previously, I found...”

£20 /hr

George H.

Degree: Chemistry (MSci) (Masters) - Bristol University

Subjects offered: Maths, Science+ 2 more

Maths
Science
Physics
Chemistry

“About Me Hi, I'm George, a chemistry student at the University of Bristol. I greatly enjoyed chemistry, maths, and physics A-levels and GCSEs, and have grown to enjoy even them more since! My passion for and knowledge of these subject...”

£20 /hr

James T.

Degree: Mathematics with Study in Continental Europe (Masters) - Bristol University

Subjects offered: Maths, German

Maths
German

“Grammar school educated tutor studying Mathematics at the University of Bristol with experience in tutoring in Maths and German”

About the author

Liam D.

Currently unavailable: no new students

Degree: Mathematics (Masters) - Queen's, Belfast University

Subjects offered: Maths

Maths

“Second year Mathematics Msc undergraduate. Looking to tutor both GCSE and A-Level Mathematics”

MyTutor guarantee

You may also like...

Posts by Liam

How do I expand brackets by multiplication?

How do I prove that an irrational number is indeed irrational?

How do I rationalise the denominator of a fraction which consists of surds?

Other A Level Maths questions

What are the rules for decomposition of partial fractions?

Statistics: Dave throws a ball at a bucket. The probability the ball goes into the bucket is 0.4. Dave throws the ball four times. What is the probability that he gets it in twice?

How do you integrate (sinx)^2?

What is the tangent line to the curve y = x^3+4x+5 at the point where x = 2?

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