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
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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

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Rationalising Denominators - Part 2
More complex surds in the denominator

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Rationalise the Denominator: Practice Questions

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Rationalising (or rationalizing) the denominator

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

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