Master Surds Rules and Examples for GCSE & A-Level

What Are Surds?

Surds are mathematical expressions containing square roots, cube roots, or other roots that cannot be simplified into a whole number or a simple fraction. For example, √2, √3, and √5 are surds because their decimal expansions are non-terminating and non-repeating.

In GCSE and A-Level Maths, mastering surds is essential for simplifying expressions, solving equations, and working with algebraic fractions.

Key Surds Rules

1. Multiplication Rule

√a × √b = √(a × b)

Example:

√2 × √3 = √(2 × 3) = √6

2. Division Rule

√a ÷ √b = √(a ÷ b)

Example:

√8 ÷ √2 = √(8 ÷ 2) = √4 = 2

3. Simplification Rule

√(a × b) = √a × √b

Example:

√50 = √(25 × 2) = √25 × √2 = 5√2

4. Addition and Subtraction of Like Surds

Only like surds can be added or subtracted, similar to algebraic terms.

Example:

3√2 + 5√2 = 8√2

4√3 − 2√3 = 2√3

5. Rationalising the Denominator

To rationalise a denominator containing a surd, multiply both the numerator and denominator by the surd.

Example:

1 ÷ √2

Multiply by √2:

(1 × √2) ÷ (√2 × √2) = √2 ÷ 2

Why Are Surds Important in Exams?

Surds often appear in GCSE and A-Level Maths exams in questions related to simplifying expressions, solving equations, or working with trigonometric identities. Understanding surds rules ensures you can approach these questions with confidence.

Step-by-Step Examples

Example 1: Simplify √72

Step 1: Break into factors:

√72 = √(36 × 2)

Step 2: Simplify:

√36 × √2 = 6√2

Final answer: 6√2

Example 2: Rationalise 5 ÷ √3

Step 1: Multiply numerator and denominator by √3:

(5 × √3) ÷ (√3 × √3) = 5√3 ÷ 3

Final answer: 5√3 ÷ 3

Example 3: Simplify (√3 + √5)(√3 − √5)

Step 1: Apply the difference of squares:

(√3)² − (√5)² = 3 − 5 = −2

Final answer: −2

Practice Exercises

Try solving these questions to test your understanding:

1. Simplify √48
2. Rationalise 7 ÷ √5
3. Simplify (√2 + √7)(√2 − √7)
4. Add 3√3 + 4√3 − √3
5. Simplify √18 ÷ √2

Check your answers with your teacher or AI tutor [LINK:/genies].

Exam Technique Tips

• Read the question carefully: Look out for keywords like "simplify," "rationalise," or "express."
• Don’t skip intermediate steps: Writing out each step ensures you don’t make careless errors.
• Memorise key rules: Familiarity with surds rules saves time during the exam.
• Use your calculator wisely: GCSE exams allow calculators, but knowing how to simplify surds manually is crucial for full marks.
• Practise past papers: Regular practice helps you spot common surd question patterns.

For more revision tips, check out our lessons [LINK:/lessons].

Conclusion

Surds are a fundamental topic in GCSE and A-Level Maths, requiring a solid understanding of their rules and applications. With regular practice and careful attention to exam techniques, you can master surds confidently.

Need extra help? Our AI tutors [LINK:/genies] are here to guide you step-by-step!