Rationalise the Denominator

Rationalizing the Denominator
It’s common practice in mathematics to express fractions without surds in the denominator. The goal is to rewrite the expression so that the denominator contains only rational numbers. This process is called rationalizing the denominator.

The following diagram shows examples of how to rationalize the denominator. Scroll down the page for more examples and solutions on how to rationalize the denominator.

 

Algebra Worksheets
Practice your skills with the following Algebra worksheets:
Printable & Online Algebra Worksheets

Case 1: Single Surd in the Denominator
Multiply both the numerator and the denominator by the surd in the denominator.
\(\frac{\sqrt{a}}{\sqrt{b}}=\frac{\sqrt{a}}{\sqrt{b}}\times \frac{\sqrt{b}}{\sqrt{b}}=\frac{\sqrt{ab}}{b}\)

Example: Rationalize \(\frac{2}{\sqrt{3}}\)
\(\frac{2}{\sqrt{3}}=\frac{2}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)

Example: Rationalize \(\frac{5}{3\sqrt{7}}\)
\(\frac{5}{3\sqrt{7}}=\frac{5}{3\sqrt{7}}\times \frac{\sqrt{7}}{\sqrt{7}}=\frac{5\sqrt{7}}{3\times 7}=\frac{5\sqrt{7}}{21}\)

Case 2: Binomial Surd in the Denominator (Using the Conjugate)
If the denominator is in the form \((a+ \sqrt{b})\) or \((a- \sqrt{b})\), multiply both the numerator and the denominator by its conjugate.
The conjugate is formed by changing the sign between the two terms.

The conjugate of \((a+ \sqrt{b})\) is \((a- \sqrt{b})\)).
The conjugate of \((a- \sqrt{b})\) is \((a + \sqrt{b})\)).
The conjugate of \((\sqrt{a} + \sqrt{b})\) is \((\sqrt{a} - \sqrt{b})\)).
The conjugate of \((\sqrt{a} - \sqrt{b})\) is \((\sqrt{a} + \sqrt{b})\)).

This uses the difference of squares identity:
(x + y)(x - y) = x² - y²
\((a+ \sqrt{b})\)\((a- \sqrt{b})= a^2 - (\sqrt{b})^2 = a^2 - b\)

Example: Rationalize \(\frac{1}{2+\sqrt{3}}\)
The conjugate of \((2 + \sqrt{3})\) is \((2 - \sqrt{3})\)
\(\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{2^{2}-(\sqrt{3})^2}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}\)

Videos

Rationalising Denominators - Part 1
A single surd in the denominator

• Show Step-by-step Solutions

Rationalising Denominators - Part 2
More complex surds in the denominator

• Show Step-by-step Solutions

Rationalise the Denominator: Practice Questions

• Show Step-by-step Solutions

Rationalising (or rationalizing) the denominator

• Show Step-by-step Solutions

How to do Surds GCSE A/A* Maths revision Higher level exam questions (simplifying, rationalising)?

• Show Step-by-step Solutions

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.