Manipulation of Surds

Q: When do I need to leave answers in surd form?

If an exam question asks for an exact answer or explicitly requests a surd, you must use surds instead of rounded decimal approximations, even if your calculator gives a decimal value. Simplify all surds fully for full marks.

Edexcel International GCSE Further Pure Mathematics· Section 1C· 15 min read

1. Surds and Exact Answers★☆☆☆☆⏱ 4 min

Surds are irrational roots of positive integers that cannot be simplified to remove their radical sign. Unlike rounded decimals, surds give fully exact values, which are required for many 4PM1 exam questions to avoid mark loss from rounding errors.

📘 Definition

Surd

An irrational number of the form $\sqrt a$ where $a$ is a positive integer with no perfect square factors other than 1.

Example:

√2 is a surd; √4 = 2 is not a surd, as it simplifies to an integer.

📐 Worked Example

Identify which of the following are surds: √9, √12, √25, √7

1
1. Simplify each radical fully to check for a remaining irrational part:
2
$√9 = 3, \text{ integer, not a surd}$
3
$√12 = √{4 \times 3} = 2√3, \text{ irrational, is a surd}$
4
$√25 = 5, \text{ integer, not a surd}$
5
$√7, \text{ no perfect square factors, is a surd}$
6
1. Final answer: √12 and √7 are surds.

Exam tip:

If a question says 'give your answer as a surd' or 'give your exact answer', you must not use a rounded decimal even if your calculator calculates the root for you.

2. Simplifying Surds and Collecting Like Terms★★☆☆☆⏱ 5 min

To simplify a surd, factor the number under the radical into the product of the largest possible perfect square and another integer. You can then take the square root of the perfect square outside the radical. Like surds have the same simplified radical part, and can be added or subtracted just like like algebraic terms.

Simplification rule: $\sqrt ab = \sqrt a \times \sqrt b$ where $a, b \geq 0$
Addition/subtraction rule: $p \sqrt x + q \sqrt x = (p + q) \sqrt x$ for like surds $\sqrt x$

📐 Worked Example

Simplify √48 + √27 - √12 fully.

1
1. Simplify each surd individually first to identify like terms:
2
$√48 = √{16 \times 3} = √16 \times √3 = 4√3$
3
$√27 = √{9 \times 3} = 3√3$
4
$√12 = √{4 \times 3} = 2√3$
5
1. All terms are now like surds with radical √3, combine their coefficients:
6
$4√3 + 3√3 - 2√3 = (4 + 3 - 2)√3 = 5√3$
7
1. Final simplified answer: $5\sqrt3$

3. Simple Products and Quotients of Surds★★★☆☆⏱ 6 min

✓ Calculator OK

Products and quotients of surds follow standard radical multiplication and division rules, and can be simplified before or after computation to get the most compact exact result. Remember that multiplying two identical surds gives an integer: $(\sqrt x)^{2} = x$ .

Product rule: $\sqrt a \times \sqrt b = \sqrt ab$
Quotient rule: $\frac{\sqrt a}{\sqrt b} = \sqrt \frac{a}{b}$
Mixed product: $p \sqrt a \times q \sqrt b = pq \sqrt ab$

📐 Worked Example

Calculate the exact value of $(3\sqrt2 \times 2\sqrt5) \div \sqrt10$ .

1
1. Compute the product first, grouping coefficients and surds separately:
2
$3√2 \times 2√5 = (3 \times 2) \times (√2 \times √5) = 6√10$
3
1. Divide by √10 using the quotient rule:
4
$\frac{6√10}{√10} = 6$
5
1. Final exact answer: 6

✓ Quick check

Test your understanding before moving on:

Simplify √18 + √8
- 5√2
- √26
- 10
- 2√5

Exam tip:

For geometry problems asking for exact area or length, use surd operations to avoid rounding and always simplify your final answer fully to score maximum marks.

4. Common Pitfalls

Wrong move:

Adding numbers under the radical when adding surds, e.g. √2 + √3 = √5

Why:

Radicals with different bases are unlike terms and cannot be combined by adding values inside the square root.

Correct move:

Leave unlike surds as separate terms, or simplify each first to check if they are like surds before combining coefficients.

Wrong move:

Forgetting to take the square root of perfect square factors when simplifying, e.g. √48 = √16×3 = 16√3

Why:

The perfect square factor's square root moves outside the radical, not the original perfect square value.

Correct move:

Take the root of the perfect square: √16 = 4, so √48 = 4√3.

Wrong move:

Using a rounded decimal approximation when an exact answer is required, e.g. writing 1.414 instead of √2

Why:

Rounded values introduce errors, and exam questions explicitly requiring exact answers will penalise decimal submissions.

Correct move:

Use fully simplified surd form for all exact answer requests, even if your calculator gives a decimal equivalent.

Wrong move:

Incorrectly adding coefficients when multiplying surd terms, e.g. 2√3 × 3√2 = 5√6

Why:

Coefficients are multiplied, not added, when calculating products of surd terms.

Correct move:

Multiply coefficients together and radicals together: (2×3)×(√3×√2) = 6√6.

Wrong move:

Attempting to simplify surds with negative values under the radical, e.g. √(-4) = 2

Why:

Real surds only exist for non-negative values under the square root, as no real number squared gives a negative result.

Correct move:

Only simplify surds with positive values under the radical, as specified in the 4PM1 syllabus.

5. Quick Reference Cheatsheet

Operation	Rule	Example
Simplification	$\sqrt ab = \sqrt a \times \sqrt b$ (a,b ≥ 0)	$\sqrt72 = \sqrt 36 \times 2 = 6\sqrt2$
Add/Subtract Like Surds	$p \sqrt x \pm q \sqrt x = (p \pm q) \sqrt x$	$5\sqrt3 - 2\sqrt3 = 3\sqrt3$
Product of Surds	$p \sqrt a \times q \sqrt b = pq \sqrt ab$	$2\sqrt5 \times 3\sqrt2 = 6\sqrt10$
Quotient of Surds	$\frac{p \sqrt a}{q \sqrt b} = \frac{p}{q} \sqrt \frac{a}{b}$	$\frac{6\sqrt15}{3\sqrt5} = 2\sqrt3$
Exact Answer Requirement	No rounded decimals, use simplified surds	Write $\sqrt2$ instead of 1.41

6. Frequently Asked

When do I need to leave answers in surd form?

If an exam question asks for an exact answer or explicitly requests a surd, you must use surds instead of rounded decimal approximations, even if your calculator gives a decimal value. Simplify all surds fully for full marks.

Can I combine √3 and √5 into a single term?

No, these are unlike surds with different radical parts, so they cannot be added or subtracted to form a single surd term. Leave them as separate terms in your final answer.

Going deeper

specificationEdexcel 4PM1 Official SpecificationSection S1 surd manipulation reference

What's Next

Now that you have mastered basic surd manipulation, you are ready to move on to rationalising denominators of surd fractions, the next sub-topic in the indices and logarithms unit of Edexcel IGCSE Further Pure Math. Surds are used extensively across the 4PM1 syllabus for exact calculations in coordinate geometry, trigonometry, and calculus problems, so practicing these foundational skills will help you avoid avoidable mark loss in later topics. Always remember to simplify all surd answers fully when an exact value is requested, and never use rounded decimals unless explicitly permitted by the question.