Surds

What Is a Surd?

A surd is a square root (or other root) whose value is irrational — it cannot be expressed as a whole number or a simple fraction. Surds allow us to write exact answers without rounding.

• √9 = 3 → not a surd (it simplifies to a whole number)
• √20 = 4.472… → surd (the decimal never ends or repeats)

Surds appear in geometry (diagonal of a square with side 1 has length √2), trigonometry (sin 60° = √3/2), and throughout algebra. Mastering them means you can carry exact values through a calculation instead of introducing rounding errors.

Simplifying Surds

To simplify a surd, find the largest perfect-square factor of the number under the root sign.

Rule: √(a × b) = √a × √b

Example 1: Simplify √72

1. Find the largest perfect-square factor of 72: 36 × 2 = 72
2. √72 = √(36 × 2) = √36 × √2 = 6√2

Example 2: Simplify √75

1. Largest perfect-square factor: 25 × 3 = 75
2. √75 = √(25 × 3) = √25 × √3 = 5√3

Tip: If you pick a smaller factor at first (e.g. √72 = √(4 × 18) = 2√18), you must simplify again until no perfect-square factor remains.

Adding and Subtracting Surds

Surds behave like algebraic terms — you can only add or subtract like surds (those with the same number under the root).

• 3√5 + 2√5 = 5√5 ✔
• 4√3 − √3 = 3√3 ✔
• 2√3 + 5√2 → cannot be combined (different surds)

Sometimes you need to simplify first before you can see which terms are alike:

3√12 + √75 = 3(2√3) + 5√3 = 6√3 + 5√3 = 11√3

Multiplying Surds

Use √a × √b = √(ab), then simplify.

Example 1: √5 × √3 = √15

Example 2: 2√3 × 4√5 = 8√15

Example 3: √6 × √6 = √36 = 6 (A surd multiplied by itself always gives a rational number — this is the key idea behind rationalising.)

Squaring a surd expression: (3√2)² = 9 × 2 = 18

Expanding Brackets with Surds

Use the same expansion rules as algebra:

Example: (2 + √3)(4 − √3)

= 2 × 4 + 2 × (−√3) + √3 × 4 + √3 × (−√3) = 8 − 2√3 + 4√3 − 3 = 5 + 2√3

The special case (a + √b)(a − √b) = a² − b produces no surds at all — this pair is called a conjugate pair and is central to rationalising.

Rationalising the Denominator

A fraction with a surd in the denominator is not in simplest form. Rationalising means rewriting it so the denominator is rational.

Case 1 — Single surd in the denominator

Multiply top and bottom by that surd:

4/√5 = (4 × √5)/(√5 × √5) = 4√5/5

Example: Simplify 10/√2

= (10 × √2)/(√2 × √2) = 10√2/2 = 5√2

Case 2 — Denominator is a + √b or a − √b

Multiply by the conjugate (change the sign between the two terms):

3/(2 + √5) × (2 − √5)/(2 − √5) = 3(2 − √5)/(4 − 5) = (6 − 3√5)/(−1) = −6 + 3√5

The denominator becomes a² − b, which is always rational.

Key Rules Summary

Operation	Rule	Example
Simplify	√(a²b) = a√b	√50 = 5√2
Multiply	√a × √b = √(ab)	√3 × √7 = √21
Square	(√a)² = a	(√11)² = 11
Add/subtract	Combine like surds only	2√5 + 3√5 = 5√5
Rationalise (simple)	Multiply by √b/√b	1/√3 = √3/3
Rationalise (conjugate)	Multiply by (a−√b)/(a−√b)	1/(1+√2) = 1−√2

Common Mistakes

• Not using the largest perfect-square factor: √48 = 4√3, not 2√12 (which is still unsimplified).
• Adding unlike surds: 3√2 + 5√3 ≠ 8√5. These terms cannot be combined.
• Forgetting to simplify before combining: Always simplify each surd first.
• Leaving a surd in the denominator: In most exams and formal answers, rationalise the denominator.
• Distributing the root incorrectly: √(a + b) ≠ √a + √b. For example, √(9 + 16) = √25 = 5, not 3 + 4.

Why Surds Matter

Exact surd form is preferred over decimals whenever precision matters. In construction, if a diagonal brace must be exactly √50 metres, writing 7.07 m introduces a rounding error that compounds across multiple measurements. Surds are also essential in trigonometric exact values (sin 45° = √2/2, cos 30° = √3/2) and in proving results in pure mathematics where approximations are not acceptable.