Surds –

Surds are irrational numbers left in root form because they cannot be written as exact decimals or fractions. They appear exclusively on Higher tier GCSE Maths papers, but mastering them unlocks marks in algebra, Pythagoras, trigonometry, and coordinate geometry. This page explains what surds are, how to simplify them, how to perform arithmetic with them, and — crucially — how to rationalise the denominator, a skill examiners test repeatedly. You will find worked examples, common mistakes, and practice questions below. For an overview of every topic you need to cover, visit our complete GCSE Maths topics list.

What Is a Surd?

A surd is a root that cannot be simplified to a whole number or a fraction. For example, √2, √3, and √5 are surds because their decimal expansions go on forever without repeating. However, √9 is NOT a surd because it equals exactly 3.

We keep surds in root form to maintain exact values. Writing √2 ≈ 1.414 loses accuracy, and in multi-step calculations this rounding error can accumulate.

Key Rules

Simplifying Surds

To simplify a surd like √72, find the largest perfect square factor: 72 = 36 × 2, so √72 = √36 × √2 = 6√2.

Always look for the largest square factor to simplify in one step rather than multiple steps.

Step-by-Step Method

Simplifying a Surd

1. Find the largest perfect square that divides into the number under the root.
2. Rewrite the surd as the product of that square root and the remaining surd.
3. Evaluate the perfect square root.

Adding and Subtracting Surds

1. Simplify each surd fully first.
2. Combine surds that have the same number under the root sign (like terms).
3. You cannot combine surds with different numbers under the root. For instance, 2√3 + 5√2 cannot be simplified further.

Multiplying Surds

1. Multiply the numbers outside the roots together.
2. Multiply the numbers under the roots together.
3. Simplify the result if possible.

Rationalising the Denominator

Simple case: denominator is √a

1. Multiply the numerator and denominator by √a.
2. The denominator becomes √a × √a = a (a rational number).
3. Simplify if possible.

Harder case: denominator is (a + √b) or (a − √b)

1. Multiply numerator and denominator by the conjugate — change the sign between the terms.
2. The denominator becomes a² − b (difference of two squares), which is rational.
3. Expand the numerator and simplify.

Worked Example 1 — Higher Level

Question: Simplify √200 + √50.

Working:

Step 1 — Simplify √200: 200 = 100 × 2, so √200 = √100 × √2 = 10√2.

Step 2 — Simplify √50: 50 = 25 × 2, so √50 = √25 × √2 = 5√2.

Step 3 — Add like surds: 10√2 + 5√2 = 15√2.

Answer: 15√2

Worked Example 2 — Higher Level

Question: Rationalise the denominator of 6 / (3 + √5). Give your answer in the form a + b√5.

Working:

Step 1 — Multiply top and bottom by the conjugate (3 − √5):

6(3 − √5) / [(3 + √5)(3 − √5)]

Step 2 — Expand the denominator using difference of two squares: (3)² − (√5)² = 9 − 5 = 4

Step 3 — Expand the numerator: 6(3 − √5) = 18 − 6√5

Step 4 — Divide through: (18 − 6√5) / 4 = 18/4 − 6√5/4 = 9/2 − 3√5/2

Answer: 9/2 − (3/2)√5 or equivalently (9 − 3√5)/2

Common Mistakes

• Trying to add unlike surds. √3 + √5 ≠ √8. You can only add surds when the numbers under the root are identical.
• Not fully simplifying. Writing √50 instead of 5√2 will cost you marks. Always check for the largest perfect square factor.
• Forgetting to multiply BOTH numerator and denominator when rationalising. You must multiply top and bottom by the same expression to keep the fraction equivalent.
• Using the wrong conjugate. For a denominator of (3 + √5), the conjugate is (3 − √5), not (−3 + √5) or (−3 − √5).
• Leaving a surd in the denominator when the question says "rationalise". If the denominator still contains a root, you have not finished.

Exam Tips

• Simplify surds as early as possible in a multi-step problem. Smaller numbers reduce calculation errors.
• Know your perfect squares up to 225. This makes it quick to spot the largest square factor. See our formulas guide for values worth memorising.
• Surds often combine with Pythagoras and trigonometry. If a right-angled triangle has sides of length √3 and √5, the hypotenuse is √(3 + 5) = √8 = 2√2. Leaving answers in surd form is usually expected.
• Check the required form. If a question says "give your answer in the form a + b√c", make sure you present your answer exactly that way.

Practice Questions

Q1: Simplify √128.

Q2: Simplify (3√2)².

Q3: Rationalise the denominator of 10/√5. Give your answer in simplified surd form.

---

Tackle Higher tier surd questions with confidence. Join GCSEMathsAI for free and get AI-powered practice matched to your ability.

Related Topics

• Powers and Roots
• Indices and Index Laws
• Expanding Brackets
• Simplifying Expressions

Summary

• A surd is a root that cannot be expressed as an exact whole number or fraction.
• Use √a × √b = √(ab) and √a × √a = a as core rules.
• To simplify, find the largest perfect square factor under the root.
• Only add or subtract surds with the same value under the root sign.
• Rationalise simple denominators by multiplying top and bottom by the surd.
• Rationalise complex denominators by multiplying by the conjugate.
• Always present your answer in the form the question requests.