How to Simplify Surds: GCSE Maths

What is a surd?

A surd is an irrational number expressed as a root. √4 = 2 exactly, so it is not a surd. √5 cannot be written exactly as a decimal or fraction — it is a surd. Surds appear throughout GCSE Higher and are essential for exact answers in trigonometry and Pythagoras questions.

Key rule: √(a × b) = √a × √b. This lets you split a root into factors.

How do you simplify a surd?

Step-by-step method:

1. Find the largest perfect square that is a factor of the number under the root.
2. Split the surd using √(a × b) = √a × √b.
3. Evaluate the square root of the perfect-square part.
4. Write the simplified form.

Worked example 1 — simplify √72:

1. Factors of 72 that are perfect squares: 4, 9, 36. Largest = 36.
2. √72 = √(36 × 2) = √36 × √2.
3. √36 = 6.
4. Answer: 6√2.

Worked example 2 — simplify √200:

1. Largest perfect-square factor of 200 = 100 (since 200 = 100 × 2).
2. √200 = √100 × √2 = 10 × √2.
3. Answer: 10√2.

How do you add and subtract surds?

You can only add or subtract like surds — those with the same number under the root sign, just as you can only add like terms in algebra.

Worked example — simplify 3√5 + 7√5 − √5:

1. All three terms have √5, so they are like surds.
2. (3 + 7 − 1)√5 = 9√5.

Worked example — simplify √12 + √27:

1. Simplify each surd first: √12 = 2√3; √27 = 3√3.
2. 2√3 + 3√3 = 5√3.

How do you multiply surds?

Use √a × √b = √(ab), then simplify if possible.

Worked example — simplify √6 × √15:

1. √6 × √15 = √(6 × 15) = √90.
2. Largest perfect-square factor of 90 = 9. √90 = √(9 × 10) = 3√10.
3. Answer: 3√10.

Note: (√a)² = a. For example, (√7)² = 7. This is used to rationalise denominators.

How do you rationalise the denominator?

A fraction with a surd in the denominator (e.g. 5/√3) needs to be rewritten so the denominator is rational. Multiply the top and bottom by the surd in the denominator.

Worked example — rationalise 5/√3:

1. Multiply numerator and denominator by √3: (5 × √3) / (√3 × √3).
2. Denominator: √3 × √3 = 3.
3. Answer: 5√3 / 3.

Worked example — rationalise 6/(2√5):

1. Multiply top and bottom by √5: (6√5) / (2 × 5) = 6√5 / 10.
2. Simplify: 3√5 / 5.

Frequently asked questions

Why is it called a surd?

The word comes from the Latin surdus meaning "deaf" or "mute" — historically, irrational roots were considered "inaudible" to the rational number system. In modern GCSE maths, a surd simply means a root that cannot be expressed exactly as a rational number.

What if I use a smaller square factor instead of the largest?

You will still reach the correct answer, but you will need extra steps. For example, simplifying √72 as √(4 × 18) = 2√18, then noticing √18 = √(9 × 2) = 3√2, giving 2 × 3√2 = 6√2. Using the largest factor (36) does it in one step. In an exam, either route is valid.

Why must the denominator be rational in a final answer?

Convention in GCSE maths (and wider mathematics) requires "simplest form" — a surd in the denominator is considered unsimplified. More practically, it is easier to compare and calculate with fractions when the denominator is a whole number.

Are surds only square roots?

No — cube roots (∛) and higher roots can also be surds if they are irrational. However, GCSE maths focuses almost exclusively on square-root surds. At A-level and beyond you encounter cube-root surds and surd-like expressions involving nth roots.

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