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Rationalising

Mathematicians dislike leaving a surd on the bottom of a fraction. Rationalising the denominator rewrites the fraction so the bottom is a whole number — without changing its value. The trick is multiplying by a cleverly chosen form of 1.

25 min Video by Zeeshan Zamurred Algebraic Expressions

Edexcel AS Maths: 1.6 Rationalising DenominatorsWatch the full walkthrough before the notes below.

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What you'll be able to do

Explain what it means to rationalise a denominator
Rationalise a denominator of the form √a
Find and use the conjugate of a two-term denominator
Rationalise denominators of the form a + √b
Leave answers in fully simplified exact form

Why rationalise?

A fraction such as $\frac{1}{2}$ has an irrational denominator. Rationalising rewrites it with a rational (whole-number) denominator, giving a tidier, standard exact form.

The key idea: multiplying the top and bottom by the same thing does not change the value of a fraction — you are multiplying by $1$ in disguise.

Rationalising a single-surd denominator

When the denominator is a single surd $a$ , multiply the top and bottom by $a$ . Since $a \times a = a$ , the surd on the bottom disappears.

\frac{1}{a} = \frac{1}{a} \times \frac{a}{a} = \frac{a}{a}

Multiply top and bottom by the surd to clear it.

1Multiply top and bottom by

3

\frac{6 3}{3}

2Simplify the numbers:

\frac{6}{3} = 2

Answer

23

Tip — After rationalising, always check whether the fraction simplifies — here 6/3 cancelled to 2.

The conjugate

When the denominator has $two$ terms, such as $a + b$ , multiply by its $conjugate$ — the same expression with the sign in the middle reversed. This creates a difference of two squares, which removes the surd.

(a + b) (a - b) = a^{2} - b

The conjugate turns a two-term surd denominator into a whole number.

1The conjugate just flips the middle sign:

3 + 5

2Product:

(3 - 5) (3 + 5) = 3^{2} - 5 = 4

Answerconjugate

3 + 5

, product

4

Rationalising a two-term denominator

Multiply the top and bottom by the conjugate of the denominator, expand carefully, and simplify. The denominator becomes rational every time.

1Multiply top and bottom by the conjugate

2 - 3

2Denominator:

(2 + 3) (2 - 3) = 4 - 3 = 1

3Numerator:

1 \times (2 - 3) = 2 - 3

Answer

2 - 3

Tip — Reverse only the MIDDLE sign for the conjugate. Everything else stays the same.

Formula recap

\frac{1}{a} = \frac{a}{a}

Single surd — multiply by √a.

(a + b) (a - b) = a^{2} - b

Conjugate clears a two-term surd.

\frac{1}{a + b} = \frac{a - b}{a ^{2} - b}

Two-term denominator result.

a \times a = a

The fact that makes it all work.

Common mistakes to avoid

Multiplying only the denominator by the surd, not the numerator.

Multiply top AND bottom by the same thing — otherwise you change the value.

Using the wrong conjugate, e.g. conjugate of 2 + √3 written as −2 + √3.

Flip only the middle sign: the conjugate of 2 + √3 is 2 − √3.

Forgetting to simplify the final fraction.

Always cancel common factors at the end, e.g. 6√3 / 3 = 2√3.

Key takeaways

Rationalising removes a surd from the denominator without changing the value.
Single surd: multiply top and bottom by that surd.
Two-term denominator: multiply by the conjugate (flip the middle sign).
The conjugate creates a difference of two squares → a rational denominator.

Test yourself

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