How to Solve: Surds (Simplifying & Rationalising the Denominator)

How to Solve: Surds (Simplifying & Rationalising the Denominator)

For GCSE & A-Level Maths (Edexcel/AQA/OCR)

Introduction

"Mastering surds doesn’t just boost your algebra grade—it’s the key to unlocking 5-10% of your GCSE/A-Level maths exam marks. One rationalised denominator question could be the difference between a 6 and a 7, or a B and an A. Let’s make sure you never lose marks on surds again."

What You Need To Know First

1. Square numbers & roots – You must instantly recognise perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) and their roots.
2. Multiplying brackets – You need to expand expressions like (a + b)(c + d) without mistakes.
3. Fraction rules – You should know how to multiply, divide, and simplify fractions.

If you’re shaky on any of these, pause here and review them first.

Key Vocabulary

Formulas To Know

1. Simplifying surds
2. √(a × b) = √a × √b (Split the root into two parts)

3. √(a² × b) = a√b (Take the square root of the perfect square)

4. Rationalising the denominator (single surd)

5. 1/√a = √a / a (Multiply numerator and denominator by √a)

6. Rationalising the denominator (binomial surd)

7. 1/(a + √b) = (a – √b) / (a² – b) (Multiply numerator and denominator by the conjugate (a – √b))
8. MEMORISE THIS: (a + √b)(a – √b) = a² – b (Difference of squares for surds)

Step-by-Step Method

Part 1: Simplifying Surds

Step 1: Factor the number under the root into a perfect square × another number. - Example: √50 → 50 = 25 × 2 (25 is a perfect square)

Step 2: Split the root using √(a × b) = √a × √b. - √50 = √(25 × 2) = √25 × √2

Step 3: Take the square root of the perfect square. - √25 = 5, so √50 = 5√2

Step 4: If possible, simplify further (e.g., 2√8 → 2 × 2√2 = 4√2).

Part 2: Rationalising the Denominator (Single Surd)

Step 1: Identify the surd in the denominator. - Example: 3/√5 → denominator is √5

Step 2: Multiply both numerator and denominator by the surd in the denominator. - 3/√5 × √5/√5 = 3√5 / 5

Step 3: Simplify if possible (here, it’s already simplified).

Part 3: Rationalising the Denominator (Binomial Surd)

Step 1: Identify the conjugate of the denominator. - Example: 1/(3 + √2) → conjugate is (3 – √2)

Step 2: Multiply both numerator and denominator by the conjugate. - 1/(3 + √2) × (3 – √2)/(3 – √2) = (3 – √2) / [(3 + √2)(3 – √2)]

Step 3: Expand the denominator using (a + b)(a – b) = a² – b². - (3 + √2)(3 – √2) = 3² – (√2)² = 9 – 2 = 7

Step 4: Write the final simplified fraction. - (3 – √2)/7

Worked Examples

Example 1 – Basic: Simplify √72

Step 1: Factor 72 into a perfect square × another number. - 72 = 36 × 2 (36 is a perfect square)

Step 2: Split the root. - √72 = √(36 × 2) = √36 × √2

Step 3: Take the square root of the perfect square. - √36 = 6, so √72 = 6√2

What we did and why: We broke 72 into 36 × 2 because 36 is the largest perfect square that divides 72. This lets us simplify √72 to 6√2, which is much cleaner.

Example 2 – Medium: Rationalise 5/(2√3)

Step 1: Identify the surd in the denominator (√3).

Step 2: Multiply numerator and denominator by √3. - 5/(2√3) × √3/√3 = 5√3 / (2 × 3)

Step 3: Simplify the denominator. - 2 × 3 = 6, so 5√3 / 6

What we did and why: We multiplied by √3/√3 (which is 1) to eliminate the surd in the denominator. This is the standard method for single-surd denominators.

Example 3 – Exam-Style: Simplify (4 + √5)/(3 – √5)

Step 1: Identify the conjugate of the denominator (3 + √5).

Step 2: Multiply numerator and denominator by the conjugate. - (4 + √5)/(3 – √5) × (3 + √5)/(3 + √5) = [(4 + √5)(3 + √5)] / [(3 – √5)(3 + √5)]

Step 3: Expand the denominator using difference of squares. - (3 – √5)(3 + √5) = 3² – (√5)² = 9 – 5 = 4

Step 4: Expand the numerator using FOIL (First, Outer, Inner, Last). - (4 + √5)(3 + √5) = 4×3 + 4×√5 + √5×3 + √5×√5 = 12 + 4√5 + 3√5 + 5 = 17 + 7√5

Step 5: Write the final simplified fraction. - (17 + 7√5)/4

What we did and why: We used the conjugate to eliminate the surd in the denominator. The numerator required careful expansion to avoid mistakes—always double-check your FOIL!

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Listen up—surds are easy marks if you follow the steps. First, simplify surds by breaking them into perfect squares. Second, rationalise single surds by multiplying top and bottom by the root. Third, for binomial denominators, multiply by the conjugate—remember, it’s just flipping the sign. Always check for hidden perfect squares, and never split roots over addition. If you see a denominator like (3 + √2), you must multiply by (3 – √2). Double-check your FOIL on the numerator, and simplify at the end. You’ve got this—go smash those surds!