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How to Simplify Surds: A Step-by-Step GCSE Maths Guide

How to Simplify Surds: A Step-by-Step GCSE Maths Guide

If you're studying for your GCSE maths exam (with AQA, Edexcel, or OCR), you've likely encountered surds. They can seem tricky at first, but with a clear method, they become much more manageable. This post will walk you through what surds are, the key rules you need to know, and how to tackle common exam questions.

What is a Surd?

A surd is an expression containing a square root (or cube root, etc.) that does not simplify to a whole number or a fraction. It is an irrational number. The most common examples you'll see are roots of non-square numbers like √2, √3, √5, and √72.

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

√5 ≈ 2.236... (this is a surd, as it cannot be written as a neat fraction).

Surds are left in their exact form because writing them as decimals would be an approximation.

The Golden Rules of Surds

Before we dive into simplification, you must know these three fundamental rules. They are the foundation for all surd manipulation.

√a × √b = √(ab)

Example: √3 × √5 = √(3×5) = √15

√a ÷ √b = √(a/b)

Example: √20 ÷ √5 = √(20/5) = √4 = 2

√(a²) = a (where a is positive)

Example: √(7²) = 7

How to Simplify a Surd: The Step-by-Step Method

Simplifying a surd means finding the largest square number that is a factor of the number inside the root.

Process:

Write the number under the square root as a product of two factors, where one factor is the largest possible square number (like 4, 9, 16, 25, 36...).

Use Rule 1 to split the root: √(a × b) = √a × √b.

Simplify the root of the square number.

Let's see this in action.

Example 1: Simplify √50.

Find the largest square factor of 50. The factors of 50 are 1, 2, 5, 10, 25, 50. The largest square number here is 25.

Rewrite: √50 = √(25 × 2)

Split the root: = √25 × √2

Simplify: = 5√2

∴ √50 simplifies to 5√2.

Example 2: Simplify √72.

Largest square factor of 72? 72 = 36 × 2. 36 is a square number.

√72 = √(36 × 2) = √36 × √2 = 6√2.

Decision Flowchart for Simplifying

Knowing when to simplify and how to start can be helped with a simple flowchart.

flowchart TD
A[Start: You have a surd √N] --> B{Is N a square number?}
B -- Yes --> C[It simplifies to an integer. Not a surd.]
B -- No --> D[Find the largest square factor Sso that N = S × T]
D --> E[Rewrite as √S × √T]
E --> F[Simplify √S to an integer]
F --> G[Final answer: integer√T]

Adding and Subtracting Surds

This is just like collecting like terms in algebra. You can only combine surds that have the same irrational part (the bit under the root).

5√3 + 2√3 = 7√3 (Like terms: both have √3).

4√2 - √2 = 3√2

√5 + 2√3 cannot be simplified further. (Unlike terms).

Crucial Tip: Always simplify each surd fully first. You might find they become like terms!

Example: Simplify √12 + √27.

Simplify individually: √12 = √(4×3) = 2√3. √27 = √(9×3) = 3√3.

Now we have: 2√3 + 3√3

Combine: = 5√3

Multiplying and Dividing Surds

Use Rules 1 and 2 directly. Remember to simplify your final answer.

Example: Simplify √8 × √6.

Multiply under one root: √(8 × 6) = √48

Now simplify the result: √48 = √(16 × 3) = 4√3.

Example: Simplify √18 ÷ √2.

Divide under one root: √(18 ÷ 2) = √9 = 3.

Rationalising the Denominator

A key GCSE skill is rationalising the denominator. This means removing a surd from the bottom (denominator) of a fraction. The rule is: multiply the top and bottom of the fraction by the surd that is in the denominator.

Case 1: Denominator is a single surd (e.g., a/√b)

Example: Simplify 5/√2.
We multiply the numerator and denominator by √2:
(5 × √2) / (√2 × √2) = (5√2) / 2

Why does this work? Because √2 × √2 = 2 (Rule 3), giving us a nice, rational denominator.

Case 2: Denominator is of the form (a + √b) or (a - √b)
Here, we use the difference of two squares. We multiply the numerator and denominator by the conjugate of the denominator. The conjugate flips the sign in the middle.

The conjugate of (a + √b) is (a - √b).

The conjugate

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