Surds Simplifying, Calculating & Rationalising the Denominator

SPEC N8 · NUMBER — SURDS

▲ HIGHER TIER ONLY

Surds

Simplifying · Calculating · Rationalising the Denominator

Exact form answers · Conjugate pairs · Paper 1 non-calculator

What is a Surd? H

SURD

An irrational root expressed in exact form, e.g. √2, √5, ∛3. The decimal expansion is non-terminating and non-recurring.

RATIONALISE

Rewrite a fraction so the denominator is rational — no surds in the denominator.

Why exact form matters

AQA questions frequently say "give your answer in simplest form" or "give your answer in exact form." This means leave it as a surd — do not use a decimal approximation. √2 ≠ 1.414.

The Four Surd Laws

Law 1

√a × √b = √(ab)

√3 × √5 = √15

Law 2

√a ÷ √b = √(a/b)

√12 ÷ √3 = √4 = 2

Law 3

(√a)² = a

(√5)² = 5 (√7)² = 7

Law 4 — Simplifying

√(a²b) = a√b (where a ≥ 0)

√(4 × 3) = √4 × √3 = 2√3. This is the key law for simplification.

⚠️ EXAMINER ALERT — surds do NOT distribute over addition.

Students write: √(9 + 16) = √9 + √16 = 3 + 4 = 7.

This is wrong: √(9 + 16) = √25 = 5, not 7.

Rule: √(a + b) ≠ √a + √b. You can only use Law 1 for multiplication, never addition.

Simplifying Surds

Method: Find the largest square factor of n, split it out using Law 4.

More Examples

√12 = √(4×3) = 2√3

√50 = √(25×2) = 5√2

√45 = √(9×5) = 3√5

Fully Simplified?

A surd is fully simplified when the number under √ has no square factors other than 1.

√8 = 2√2 ✓ (2 has no square factors)

2√8 ✗ (8 = 4×2, not fully simplified)

Adding & Subtracting Surds

Treat surds like algebraic terms — you can only collect like surds (same number under the √).

Like surds — can collect

2√3 + 5√3 = 7√3 (same as 2x + 5x = 7x)

8√5 − 3√5 = 5√5

Unlike surds — cannot collect

√2 + √3 cannot be simplified further — different radicands.

Simplify FIRST, then collect

√8 + √18 — these look unlike, but simplify first:

√8 = 2√2 | √18 = 3√2

2√2 + 3√2 = 5√2

⚠️ EXAMINER ALERT — standard AQA trap (cross-series). Students stop at √8 + √18 and say the two surds are "unlike" and "cannot be combined." They are right that √8 and √18 are unlike as written — but wrong to stop there. Always simplify each term first: √8 = 2√2 and √18 = 3√2, which are like surds. Failing to simplify before adding is one of the most common ways marks are lost on surd questions.

Practice Question

Simplify √12 + √75. Give your answer in simplest surd form. 3 marks

M:

Simplify each surd separately, then collect like terms.

C:

√12 = √(4 × 3) = 2√3

√75 = √(25 × 3) = 5√3

2√3 + 5√3 = 7√3

E:

7√3 ✓ — fully simplified (3 has no square factors).

Multiplying Surds

Multiply the integer parts together, multiply the surd parts together.

p√a × q√b = pq√(ab)

Example 1 — Basic

2√3 × 5√2

= (2 × 5) × √(3 × 2)

= 10√6 ✓

Example 2 — Simplify result

3√2 × 4√8

= 12 × √(2 × 8) = 12√16

= 12 × 4 = 48 ✓

Squaring a surd — Law 3

(√a)² = a (√a)(√a) = a

(√5)² = 5 | (√7)² = 7 | (3√2)² = 9 × 2 = 18

Exam tip

After multiplying, always check: can the surd under the √ be simplified further? √16 = 4 — that is a whole number, so 12√16 = 48. Always simplify the result.

Expanding Brackets with Surds

Use the same FOIL / grid method as algebraic expansion.

Difference of Two Squares — key pattern

(a + √b)(a − √b) = a² − b

Example: (2 + √3)(2 − √3) = 4 − 3 = 1

Example: (5 + √2)(5 − √2) = 25 − 2 = 23

This is the key identity used in rationalising with a conjugate.

General expansion

(2 + √3)² = (2 + √3)(2 + √3)

= 4 + 2√3 + 2√3 + (√3)²

= 4 + 4√3 + 3 = 7 + 4√3

Mark scheme note

Show every expansion step — examiners award M marks for correct intermediate working. Writing only the final answer loses method marks if it is wrong.

Practice Question

Work out (2 + √3)² − (2 − √3)².

Give your answer in simplest form. 3 marks

M:

Expand each bracket separately, then subtract.

C:

(2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3

(2 − √3)² = 4 − 4√3 + 3 = 7 − 4√3

Difference: (7 + 4√3) − (7 − 4√3) = 8√3

E:

8√3 ✓ — the integer parts cancel, leaving 8√3.

Rationalising — Simple Case

When the denominator is a single surd √a, multiply top and bottom by √a.

3/√2 = (3 × √2)/(√2 × √2) = 3√2/2

Example 1

6/√3 = (6√3)/(√3 × √3) = 6√3/3 = 2√3

Example 2

10/√5 = (10√5)/(√5 × √5) = 10√5/5 = 2√5

Rationalising — Conjugate Case

When the denominator has the form (a + b√c) or (a − b√c), use the conjugate — flip only the sign between the two terms.

Conjugate pairs

(3 + √2) and (3 − √2) are conjugates.

(√7 − √3) and (√7 + √3) are conjugates.

Key result: (a + √b)(a − √b) = a² − b (no surds!)

Method

5/(3 + √2)

Multiply top and bottom by the conjugate (3 − √2):

= 5(3 − √2) / [(3 + √2)(3 − √2)]

Denominator: 3² − (√2)² = 9 − 2 = 7

= 5(3 − √2) / 7 = (15 − 5√2) / 7

⚠️ EXAMINER ALERT — wrong conjugate. Students multiply by (3 + √2) again (the same expression). The conjugate flips the sign only — (3 + √2) → (3 − √2). Multiplying by the same expression does not clear the surd from the denominator.

MCE Worked Example

Rationalise the denominator: 4 / (√7 − √3).

Give your answer in simplest form. 3 marks

M Denominator is (√7 − √3).

Conjugate is (√7 + √3).

Multiply top and bottom by (√7 + √3).

C Numerator: 4(√7 + √3)

Denominator: (√7)² − (√3)²

= 7 − 3 = 4

Fraction: 4(√7 + √3)/4

E The 4 cancels.

Answer: √7 + √3

Exact form ✓ Rationalised ✓

Mark scheme

M1: correct conjugate (√7 + √3) identified. M1: denominator expanded to (√7)² − (√3)² = 4. A1: √7 + √3 in simplest form. The cancellation of the 4 is expected and must be shown.

Practice Question

Rationalise the denominator: 5 / (3 + √2).

Give your answer in the form (a + b√2) where a and b are rational numbers. 3 marks

M:

Conjugate of (3 + √2) is (3 − √2). Multiply top and bottom by (3 − √2).

C:

Numerator: 5(3 − √2) = 15 − 5√2

Denominator: (3 + √2)(3 − √2) = 3² − (√2)² = 9 − 2 = 7

Result: (15 − 5√2) / 7

E:

(15 − 5√2) / 7 ✓

In the form (a + b√2): a = 15/7, b = −5/7. Both are rational. The denominator is rational — rationalisation is complete.

Exam-Style: Nov24 P1 Q20

Simplify (√8 + √18) / √2.

Give your answer in simplest form. 3 marks

M:

Simplify each surd in the numerator first, then divide by √2.

C:

√8 = √(4 × 2) = 2√2

√18 = √(9 × 2) = 3√2

Numerator: 2√2 + 3√2 = 5√2

(5√2) / √2 = 5

E:

5 ✓ — the result is a whole number. The surds cancelled exactly.

Note: many students in Nov24 applied rationalisation here — it was not needed. The simplest approach is to simplify the numerator first.

Practice Question

Simplify and rationalise: 6/√3 + √12.

Give your answer in simplest surd form. 3 marks

M:

Rationalise 6/√3 by multiplying top and bottom by √3. Then simplify √12. Then add.

C:

6/√3 = 6√3/(√3 × √3) = 6√3/3 = 2√3

√12 = √(4 × 3) = 2√3

2√3 + 2√3 = 4√3

E:

4√3 ✓ — rationalised and fully simplified.

Common Errors — Don't Lose These Marks

Students often writeCorrect versionError type

√(9 + 16) = √9 + √16 = 7√25 = 5Surds don't distribute over addition

√8 + √18 "cannot simplify"2√2 + 3√2 = 5√2Simplify each term first

√50 left as √505√2Not fully simplified — 25 is a square factor

6/√3 left as is2√3Must rationalise when asked for "simplest form"

Conjugate of (3 + √2) is (3 + √2)(3 − √2)Conjugate flips the sign only

Answer as 2√8 (unsimplified surd)4√2Mark scheme requires no square factors under √

Summary

Key Points — Lesson 1.09 H

Surd: irrational root in exact form. Exact form ≠ decimal.

Simplify: find the largest square factor. √72 = √(36×2) = 6√2.

Add/subtract: simplify each term first, then collect like surds.

Multiply: integers × integers, surds × surds. Always simplify the result.

(√a)² = a. DOTS: (a + √b)(a − √b) = a² − b.

Rationalise simple: multiply top and bottom by √a.

Rationalise conjugate: multiply by (a ∓ b√c) — flip the sign only.

Final answer: no square factors under √, no surds in denominator.

Must-know results

√(a + b) ≠ √a + √b (fundamental rule)

(√a)² = a

(a + √b)(a − √b) = a² − b

Conjugate of (p + q√r) is (p − q√r)

Quick simplification check

Is the number under √ divisible by a perfect square (4, 9, 16, 25, 36…)? If yes — simplify. If no — fully simplified.

Exam Tip

Every final surd answer must pass two tests: (1) no square factors remain under the √, and (2) no surds in the denominator. Fail either test and the A mark is lost even if all method marks were earned.