Surds

√4 isn't a surd, because it equals 2 exactly. √2 is a surd: 1.41421356... and the digits never end or settle into a pattern, because √2 is irrational. If you round it, you lose accuracy, so when a question asks for an exact answer, it wants the surd.

The rules of surds

Surds behave like algebra: √2 is like x. You can collect 4√2 + 5√2 just as you collect 4x + 5x, but you can't combine √2 + √3, any more than x + y.

How to simplify surds

1. Find the largest square number that divides the number under the root. The squares are 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on.
2. Split the root: √(square × other) = √square × √other.
3. Work out the square root of the square and write it in front.

Worked example: simplify √72. 72 = 36 × 2, and 36 is the largest square factor, so √72 = √36 × √2 = 6√2. If you only spot 72 = 4 × 18, you get 2√18, which still isn't simplest: √18 = 3√2, giving 2 × 3√2 = 6√2. Always check whether the number left under the root has another square factor.

Adding and subtracting surds

You can only add or subtract surds with the same number under the root, so simplify first.

On their own, √8 and √18 look unrelated. Simplified, both are multiples of √2 and collect into 5√2.

Multiplying surds

Multiply the numbers outside the roots together, and the numbers inside together.

To expand brackets, multiply every term by every term, just as in algebra, and remember √a × √a = a.

In the second one, the surds cancel and the answer is a whole number. That always happens with a pair like (a + √b)(a − √b), because it is a difference of two squares: a² − b. It is the trick behind rationalising.

Rationalising the denominator

Rationalising means rewriting a fraction so there is no surd on the bottom. It makes answers simpler to compare and is often required for full marks.

A single surd on the bottom

Multiply the top and bottom by that surd:

A bracket on the bottom

Multiply the top and bottom by the same bracket with the sign in the middle swapped, called the conjugate. The bottom becomes a difference of two squares, and the surd disappears.

Check with a calculator: 4 ÷ (3 − √5) and 3 + √5 both come to 5.236..., so the answer is right.

Surds in exam questions

In GCSE Maths, simplifying surds and rationalising denominators are Higher tier skills, part of calculating exactly with fractions, surds and multiples of π in the AQA specification. Surds also turn up inside other topics:

• Pythagoras: a right-angled triangle with shorter sides 2 cm and 4 cm has a hypotenuse of √(4 + 16) = √20 = 2√5 cm exactly.
• The quadratic formula: solving x² − 6x + 4 = 0 gives x = 3 ± √5, an exact answer in surd form.
• Trigonometry: exact values such as sin 60° = √3/2 and sin 45° = √2/2 (or 1/√2).
• Area and volume: a question asking for an answer in the form a√b wants the surd fully simplified.

Surds are also powers: √a is the same as a^½, which links them to the index laws. The Exponent Calculator works with fractional powers like these. For short timed practice on Higher number and algebra, try a daily GCSE Maths Higher paper. Our guide to completing the square shows where surd answers like 4 ± √13 come from.

Why is √2 irrational?

Suppose √2 could be written as a fraction a/b in its lowest terms. Then 2 = a²/b², so a² = 2b², which makes a² even and so a even. Write a = 2k: then 4k² = 2b², so b² = 2k², making b even too. But a and b can't both be even if the fraction was in its lowest terms. The assumption was impossible, so √2 isn't a fraction. The discovery that some lengths can't be written as fractions goes back to the ancient Greeks, according to MacTutor.

Common mistakes

• Adding unlike surds: √2 + √3 is not √5. Leave it as it is.
• Splitting a sum: √(9 + 16) is 5, not √9 + √16 = 7.
• Not simplifying fully: 2√18 should become 6√2.
• Using the wrong factor: take out a square factor, not just any factor. √12 = √4 × √3, not √6 × √2.
• Forgetting to multiply the top when rationalising: whatever you do to the bottom, do to the top.
• Rounding when an exact answer is asked for. Give the surd.

Read next

Maths · NumberSquare root: meaning, table, simplifying and examplesWhat a square root is, a table of square roots, how to find and estimate them, how to simplify square roots like √72, and the square root graph.