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Rationalising the Denominator (Surds): GCSE Revision with Examples

Rationalising the denominator is useful because it helps us simplify fractions with surds, making them easier to work with. By removing surds from the denominator, it allows for easier manipulation of algebraic expressions.

What Does Rationalising the Denominator Mean?

Rationalising the denominator means rewriting a fraction with an irrational denominator as an equivalent fraction with a rational denominator.

A rational number is one that can be written as a fraction. Surds such as √7 and √11 are irrational numbers.

When we rationalise a denominator, we want to replace the surd with a whole number.

\( For \: example, \frac{2}{\sqrt{3}} \: can \: be \: rewritten \: as \: \frac{2\sqrt{3}}{3} \)

The first fraction has an irrational denominator, √3, while the second has a rational one, 3.

If you don’t know how to do this don’t worry; in this lesson you’ll learn exactly how to rationalise the denominator.

How to Rationalise the Denominator

Rationalising the Denominator Method

Think back to when you first learned about equivalent fractions.

If you multiply the numerator and denominator by the same number, you get an equivalent fraction.

For example:

\( \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \)

3/5 and 6/10 are equivalent fractions.

When dealing with fractions with surds, we can also find equivalent fractions.

When the denominator has only one term, we can rationalise the denominator by multiplying the numerator and denominator by the same surd.

Eg.

\( \frac{2}{\sqrt{5}} = \frac{2 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{2\sqrt{5}}{5} \)

Step-by-Step Method

  1. Simplify the surds first, if possible.
  2. Multiply the numerator and denominator by the surd in the denominator.
  3. Simplify the fraction, if possible.

Step-by-Step Examples

Example 1

\( Rationalise \: \frac{6}{\sqrt{3}} \)

Step 1: √3 cannot be simplified.

Step 2: Multiply the numerator and denominator by √3.

\( \frac{6}{\sqrt{3}} = \frac{6 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} \\ = \frac{6\sqrt{3}}{3} \)

Step 3: Simplify the fraction, if possible

\( \frac{6\sqrt{3}}{3} = 2\sqrt{3} \)

Answer: 2√3

Example 2

\( Rationalise \: \frac{8}{\sqrt{24}} \)

Step 1: We can simplify first.

√24 = √4√6 = 2√6

Step 2: Multiply the numerator and denominator by √6.

\( \frac{8}{\sqrt{24}} = \frac{8}{2\sqrt{6}} \\ = \frac{8 \times \sqrt{6}}{2\sqrt{6} \times \sqrt{6}} \\ = \frac{8\sqrt{6}}{2 \times 6} \\ = \frac{8\sqrt{6}}{12} \)

Step 3: Simplify the fraction, if possible

\( \frac{8\sqrt{6}}{12} = \frac{2\sqrt{6}}{3} \)

Answer:

\( \frac{2\sqrt{6}}{3} \)

Example 3

\( Rationalise \: \frac{\sqrt{3}}{4\sqrt{5}} \)

Step 1: We cannot simplify √3 or √5.

Step 2: Multiply the numerator and denominator by √5.

\( \frac{\sqrt{3}}{4\sqrt{5}} = \frac{\sqrt{3} \times \sqrt{5}}{4\sqrt{5} \times \sqrt{5}} \\ = \frac{\sqrt{15}}{4 \times 5} \\ = \frac{\sqrt{15}}{20} \)

Step 3: This cannot be simplified.

Answer:

\( \frac{\sqrt{15}}{20} \)

Example 4

\( Rationalise \: \frac{2\sqrt{18}}{\sqrt{20}} \)

Step 1: We can simplify both surds first.

\( \frac{2\sqrt{18}}{\sqrt{20}} = \frac{2\sqrt{9}\sqrt{2}}{\sqrt{4}\sqrt{5}} \\ = \frac{2 \times 3\sqrt{2}}{2\sqrt{5}} \\ = \frac{ 6\sqrt{2}}{2\sqrt{5}} \\ = \frac{ 3\sqrt{2}}{\sqrt{5}} \)

Step 2: Multiply the numerator and denominator by √5.

\( \frac{3\sqrt{2}}{\sqrt{5}} = \frac{3 \sqrt{2} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} \\ = \frac{3\sqrt{10}}{5} \)

Step 3: This cannot be simplified.

Answer:

\( \frac{3\sqrt{10}}{5} \)

Rationalising the Denominator (with 2 Terms)

Rationalising a denominator with only 1 term is fairly straight forward but the same method will not work when we try to do this with 2 terms.

Eg.

\( \frac{2}{\sqrt{5} \: + \: 3} = \frac{2 \times \sqrt{5}}{(\sqrt{5} \: + \: 3 ) \times \sqrt{5}} \\ = \frac{2\sqrt{5}}{5 \: + \: 3\sqrt{5}} \)

In this example, multiplying by √5 does not rationalise the denominator.

How to Rationalise a Denominator with 2 Terms

Before we can rationalise a denominator with 2 terms, we need to understand what a conjugate is.

A conjugate is formed by changing the sign between two terms.

For example, the conjugate of 3 + √2 is 3 – √2 and the conjugate of 5√2 – 4 is 5√2 + 4.

Step-by-Step Method

We can rationalise a denominator with 2 terms by multiplying the numerator and denominator by the conjugate of the denominator.

  1. Simplify the surds, if possible.
  2. Multiply the numerator and denominator by the conjugate of the denominator.
  3. Simplify the fraction, if possible.

Examples

Example 1

\( Rationalise \: \frac{2}{\sqrt{5} \: + \: 3} \)

Step 1: √5 cannot be simplified.

Step 2: The conjugate of √5 + 3 is √5 – 3.

So multiply the numerator and denominator by √5 – 3.

\( \frac{2}{\sqrt{5} \: + \: 3} = \frac{2(\sqrt{5} \: – \: 3)}{(\sqrt{5} \: + \: 3)(\sqrt{5} \: – \: 3)} \\ = \frac{2\sqrt{5} \: – \: 6}{5 \: – \: 3\sqrt{5} \: + \: 3\sqrt{5} \: – \: 9} \\ = \frac{2\sqrt{5} \: – \: 6}{-4} \)

Step 3: We can simplify the fraction by taking out a factor of -2.

\( \frac{2\sqrt{5} \: – \: 6}{-4} = \frac{-\sqrt{5} \: + \: 3}{2} \)

Read our lesson on expanding brackets with surds if you didn’t understand the first step.

Example 2

\( Rationalise \: \frac{8}{\sqrt{6} \: – \: 2} \)

Step 1: √6 cannot be simplified.

Step 2: The conjugate of √6 – 2 is √6 + 2.

So multiply the numerator and denominator by √6 + 2.

\( \frac{8}{\sqrt{6} \: – \: 2} = \frac{8(\sqrt{6} \: + \: 2)}{(\sqrt{6} \: – \: 2)(\sqrt{6} \: + \: 2)} \\ = \frac{8\sqrt{6} \: + \: 16}{6 \: – \: 2\sqrt{6} \: + \: 2\sqrt{6} \: – \: 4} \\ = \frac{8\sqrt{6} \: + \: 16}{2} \)

Step 3: We can simplify the fraction by taking out a factor of 2.

\( \frac{8\sqrt{6} \: + \: 16}{2} = 4\sqrt{6} \: + \: 8 \)

Example 3

We can apply the same method when both numerator and denominator have 2 terms. Remember that we take the conjugate of the denominator only.

\( Rationalise \: \frac{\sqrt{8} \: + \ 3}{\sqrt{8} \: – \: 2} \)

Step 1: √8 simplifies to 2√2.

The fraction can be rewritten as:

\( \frac{\sqrt{8} \: + \ 3}{\sqrt{8} \: – \: 2} = \frac{2\sqrt{2} \: + \ 3}{2\sqrt{2} \: – \: 2} \)

Step 2: The conjugate of 2√2 – 2 is 2√2 + 2.

So multiply the numerator and denominator by 2√2 + 2.

\( \frac{2\sqrt{2} \: + \ 3}{2\sqrt{2} \: – \: 2} = \frac{(2\sqrt{2} \: + \ 3)(2\sqrt{2} \: + \ 2)}{(2\sqrt{2} \: – \: 2)(2\sqrt{2} \: + \: 2)} \)

Numerator:

\( (2\sqrt{2} \: + \ 3)(2\sqrt{2} \: + \ 2) \\ = 8 + 4\sqrt{2} + 6\sqrt{2} + 6 \\ = 14 + 10\sqrt{2} \)

Denominator:

\( (2\sqrt{2} \: – \: 2)(2\sqrt{2} \: + \: 2) \\ = 8 \: + \: 4\sqrt{2} \: – \: 4\sqrt{2} \: – \: 4 \\ = 4 \)

Fraction:

\( \frac{14 \: + \: 10\sqrt{2}}{4} \)

Step 3: We can simplify the fraction by taking out a factor of 2.

\( \frac{14 \: + \: 10\sqrt{2}}{4} = \frac{7 \: + \: 5\sqrt{2}}{2} \)

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