Simplifying Surds –

Simplifying surds is a key Higher tier topic that appears regularly on AQA, Edexcel, and OCR GCSE Maths papers. A surd is an irrational root that cannot be simplified to a whole number, and exam questions expect you to leave answers in exact surd form rather than as decimals.

What Is Simplifying Surds?

A surd is a root that cannot be written as an exact whole number or fraction — for example, √2, √3, and √5 are all surds. √4 is not a surd because it equals exactly 2.

Simplifying a surd means rewriting it so the number under the root sign is as small as possible. You do this by finding the largest square number that is a factor of the number under the root. For example, √48 = √(16 × 3) = √16 × √3 = 4√3.

You can also add and subtract surds that have the same number under the root sign, just like collecting like terms. And you can multiply surds using the rule √a × √b = √(ab).

Key Formulas

Step-by-Step Method

1. To simplify a surd, find the largest square factor of the number under the root.
2. Split the surd: √(largest square × remaining) = √(square) × √(remaining).
3. Take the square root of the square number out in front.
4. To add or subtract surds, simplify each surd first, then combine like terms.
5. To multiply surds, use √a × √b = √(ab), then simplify the result.

Worked Example 1 — Foundation Level

Question: Simplify √72.

Working:

Step 1 — Find the largest square factor of 72. The square numbers are 1, 4, 9, 16, 25, 36. The largest that divides 72 is 36.

Step 2 — Split: √72 = √(36 × 2) = √36 × √2.

Step 3 — √36 = 6, so √72 = 6√2.

Answer: 6√2

Worked Example 2 — Higher Level

Question: Simplify √50 + √18. Give your answer in the form a√b.

Working:

Step 1 — Simplify √50: √50 = √(25 × 2) = 5√2.

Step 2 — Simplify √18: √18 = √(9 × 2) = 3√2.

Step 3 — Both surds are now multiples of √2, so add: 5√2 + 3√2 = 8√2.

Answer: 8√2

Worked Example 3 — Exam Style

Question: Show that (2√3)² + (√5)² = 17. (2 marks)

Working:

Step 1 — (2√3)² = 2² × (√3)² = 4 × 3 = 12.

Step 2 — (√5)² = 5.

Step 3 — 12 + 5 = 17. As required.

Answer: Shown: (2√3)² + (√5)² = 12 + 5 = 17

Common Mistakes

• Not finding the largest square factor. Writing √72 = √(4 × 18) = 2√18 is not fully simplified. Always look for the largest square factor, or continue simplifying until the number under the root has no square factors left.
• Adding surds with different roots. You cannot add √2 + √3 — they are unlike terms. You can only combine surds when the number under the root is the same.
• Forgetting to square the coefficient. When squaring 3√5, both parts must be squared: (3√5)² = 9 × 5 = 45, not 3 × 5 = 15.

Exam Tips

• Memorise the square numbers up to 225 (15²) so you can spot square factors quickly in the exam.
• When simplifying a surd, if you do not immediately spot the largest square factor, you can simplify in stages: √72 = √(4 × 18) = 2√18 = 2√(9 × 2) = 2 × 3√2 = 6√2.
• "Give your answer in surd form" means you must not use a calculator to write a decimal. Leave roots in place.

Practice Questions

Q1 (Higher): Simplify √200.

Q2 (Higher): Simplify 3√12 − √27.

Q3 (Higher): Expand and simplify √2(√8 + √18).

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Related Topics

• Surds
• Indices and Index Laws
• Rationalising the Denominator

Summary

• A surd is an irrational root that cannot be written as an exact whole number.
• To simplify, find the largest square factor and take its root outside.
• You can add or subtract surds only when they share the same root (like collecting like terms).
• √a × √b = √(ab) — use this to multiply and then simplify.
• Always fully simplify so the number under the root has no square factors.