If you're studying for your GCSE maths exam (with AQA, Edexcel, or OCR), you've likely encountered surds. They are numbers left in square root form (√) that give irrational answers. Understanding them is key to securing marks in the Number and Algebra sections. This guide will walk you through the core skills you need.
What is a Surd?
A surd is an irrational number expressed as the root of an integer, most commonly a square root. For example, √2, √3, and √5 are all surds because their decimal forms go on forever without repeating. Numbers like √4 or √9 are not surds because they simplify to the integers 2 and 3.
The Golden Rule: √a × b = √a × √b
This rule is the foundation for everything you do with surds. It allows you to break down a square root into simpler parts.
Step 1: Simplifying Surds
The goal is to find the largest square number that is a factor of the number inside the root.
Method:
Find two factors of the number, where one is the largest square number you can find (like 4, 9, 16, 25, 36...).
Rewrite the surd using the rule: √square × other.
- Take the square root of the square number.
Example 1: Simplify √12.
Factors of 12: 4 and 3 (4 is a square number).
- √12 = √4 × 3 = √4 × √3 = 2√3.
Example 2: Simplify √72.
The largest square factor of 72 is 36 (36 × 2 = 72).
- √72 = √36 × 2 = √36 × √2 = 6√2.
flowchart TD
A["Start: Surd to Simplify e.g. √72"] --> B["Find Largest Square Factor, e.g. 36"];
B --> C["Rewrite as √Square × √Other, √36 × √2"];
C --> D["Simplify Square Root, 6√2"];
D --> E["End: Simplified Surd"];
Step 2: Adding and Subtracting Surds
You can only add or subtract surds if they have the same surd part (like collecting like terms in algebra).
Example: Simplify √50 + √18.
Simplify each surd first:
√50 = √25 × 2 = 5√2
√18 = √9 × 2 = 3√2
- Now add: 5√2 + 3√2 = (5+3)√2 = 8√2
You cannot simplify √2 + √3. This is already in its simplest form.
Step 3: Multiplying and Dividing Surds
Use these rules:
√a × √b = √ab
- √a ÷ √b = √a/b
Example: Simplify √6 × √10.
√6 × √10 = √6 × 10 = √60.
- Now simplify √60: √4 × 15 = 2√15.
Remember: Always simplify your final answer.
Step 4: Rationalising the Denominator
A key GCSE skill is to rewrite a fraction so the denominator is a rational number (i.e., gets rid of the surd on the bottom). This is called rationalising the denominator.
Case 1: Denominator is a single surd (e.g., a√b)
Multiply the numerator and denominator by the surd in the denominator.
Example: Rationalise 5√3.
- Multiply top and bottom by √3: (5 × √3) / (√3 × √3) = 5√33.
Case 2: Denominator is of the form (a + b√c) (e.g., 1(1+√2))
Multiply the numerator and denominator by the conjugate of the denominator. The conjugate is the same expression with the opposite sign between the terms (e.g., the conjugate of 1+√2 is 1-√2).
Example: Rationalise 1(1+√2).
Multiply top and bottom by the conjugate (1-√2).
Numerator: 1 × (1-√2) = 1 - √2.
Denominator: (1+√2)(1-√2). Expand using the difference of two squares: (1)² - (√2)² = 1 - 2 = -1.
- So the fraction becomes: 1 - √2-1 = -1 + √2 or √2 - 1.
flowchart TD
A["Fraction with Surd in Denominator"] --> B["What form is the denominator?"];
B -- Single Surd<br>e.g. /√5 --> C["Multiply top and bottom by that surd"];
B -- Two terms with surd<br>e.g. / 3+√7 --> D["Multiply top and bottom by the conjugate"];
C --> E["Simplify the fraction"];
D --> E;
E --> F["Rationalised Answer"];
Step 5: Expanding Brackets with Surds (FOIL)
This works just like expanding algebraic brackets. Use the FOIL (First, Outer, Inner, Last) method.
Example: Expand and simplify (√3 + 2)(√3 - 5).
F: √3 × √3 = 3
O: √3 × (-5) = -5√3
I: 2 × √3 = 2√3
L: 2 × (-5) = -10
- Combine: 3 + (-5√3) + (2√3) + (-10) = 3 - 10 -5√3 + 2√3 = -7 - 3√3
Common Mistakes to Avoid
Adding different surds: √2 + √3 ≠ √5. You cannot combine them.
Incorrect simplifying: √a + b ≠ √a + √b. This is a very common error! The rule only works for multiplication inside the root.
Forgetting to simplify fully: Always check if the number inside the surd has any square factors.
- Misapplying the conjugate: Only use the conjugate when rationalising a denominator with two terms.
Practice for Your Exam
Try these typical GCSE questions:
Simplify √75 + √27. (Answer: 5√3 + 3√3 = 8√3)
Rationalise the denominator of 10√5. (Answer: 2√5)
- Expand and simplify (4 - √2)(1 + 3√2). (Answer: 4 + 12√2 - √2 - 3×2 = 4 + 11√2 - 6 = -2 + 11√2)
Mastering these steps will make surds a confident part of your GCSE maths toolkit. Remember to always look for square factors and practise rationalising until it becomes second nature. Good luck!
- This content is aligned with the KS4 Maths curriculum and relevant to all major UK exam boards including AQA, Edexcel and OCR.