GCSE MATHS • HIGHER TIER

Simplifying Surds Made Simple

Recognise square-number factors and simplify irrational square roots into exact surd form.

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Simplifying Surds in Four Moves

1

Find

Find the largest square-number factor inside the surd.

2

Split

Rewrite the surd as a product of square roots.

3

Simplify

Take the square root of the square-number factor.

4

Check

Check whether the remaining surd can be simplified further.

What You’ll Learn

Understand what a surd is.

Understand what simplifying a surd means.

Use √(ab) = √a × √b.

Recognise useful square-number factors.

Simplify surds into exact form.

Understand why exact values matter.

Avoid common GCSE surd mistakes.

Key Rules

The most important surd rule is :

√(ab) = √a × √b
This allows us to split square roots into smaller parts and identify square-number factors.

Worked Proof: Simplifying Surds

Step-by-Step Method

  • 1. Find the largest square-number factor.
  • 2. Rewrite the surd as a product of square roots.
  • 3. Simplify the square root of the square number.
  • 4. Write the remaining surd inside the root sign.
  • 5. Check whether further simplification is possible.

More Worked Examples

1

Simplify √12.

Step 1 : Find the largest square factor.

12 = 4 × 3.
Step 2: Split the root.
√12 = √4 × √3.
Step 3: Simplify.
√4 = 2.
≈ 31.4 cm

2

Simplify √18.

18 = 9 × 2

√18 = √9 × √2.
= 3√2.
Answer : 3√2

3

Simplify √20.

20 = 4 × 5.

√20 = √4 × √5.
= 2√5.
Answer : 2√5

4

Simplify √45.

45 = 9 × 5.

√45 = √9 × √5.
= 3√5.
Answer : 3√5
  • Common GCSE Mistakes
  • Choosing a small square factor and not simplifying fully.
  • Taking non-square factors outside the root sign.
  • Adding or subtracting numbers incorrectly inside square roots.
  • Giving a decimal when an exact surd answer is required.
  • Exam Tips
  • Look for the largest square-number factor first.
  • Memorise common square numbers up to 100.
  • Use √(ab) = √a × √b carefully.
  • Keep answers exact unless a decimal is requested.
  • Check whether the remaining surd simplifies again.
  • Make sure the final surd is fully simplified.

Skills Used in Simplifying Surds

Surds

Square Numbers

Factors

Square Roots

Exact Values

Number Skills

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Frequently Asked Questions

A surd is a root that cannot be simplified into a whole number and is kept in root form to preserve an exact value.

It means rewriting the surd in its simplest exact form by removing square-number factors from inside the root.

√(ab) = √a × √b.

It makes the simplification faster and helps ensure the final answer is fully simplified.

Surds preserve exact values, while decimal approximations can introduce rounding errors.

The source recommends 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.