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.

LD
Answered by Liam D. Maths tutor

9338 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

A circle A has equation x^2+y^2-6x-14y+54=0. Find a) the coordinates of the centre of A, b) the radius of the circle A.


Find the exact solution, in its simplest form, to the equation ln(4y + 7) = 3 + ln(2 – y) (Core Maths 3 Style Question)


What is a logarithm?


Simplify: 3l^2mn+nl^2m−5mn^2l+l^2nm+2n^2ml−mn^2


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences