GCSE Maths Number - How to Solve: Surds (Simplifying & Rationalising the Denominator) – Complete Guide

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

Introduction

"Mastering surds unlocks 5–10 marks in your GCSE/A-Level Maths exam—plus the confidence to tackle physics equations with square roots in resistors, half-lives, and wave speeds. One wrong simplification, and your answer is worth zero. Today, you’ll learn the exact steps to simplify surds and rationalise denominators—no guesswork, just full marks."

WHAT YOU NEED TO KNOW FIRST

1. Square numbers: You must recognise perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225).
2. Prime factorisation: Breaking numbers into products of primes (e.g., 18 = 2 × 3 × 3).
3. Fraction rules: Multiplying numerators and denominators by the same number doesn’t change the value.

KEY TERMS & FORMULAS

Key Terms

• Surd: A root (√) that cannot be simplified to a whole number (e.g., √2, √3, √12).
• Rationalising the denominator: Removing the surd from the denominator of a fraction.
• Conjugate: For an expression like a + √b, its conjugate is a – √b (and vice versa).

Formulas

1. Simplifying surds:
   [
   \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
   ]
2. a and b are positive integers.

3. MEMORISE THIS: Use it to split surds into simpler parts.

4. Rationalising the denominator (single surd):
   [
   \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}
   ]

5. Multiply numerator and denominator by √a.

6. MEMORISE THIS: The denominator becomes a whole number.

7. Rationalising the denominator (binomial surd):
   [
   \frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{(a + \sqrt{b})(a - \sqrt{b})} = \frac{a - \sqrt{b}}{a^2 - b}
   ]

8. Multiply numerator and denominator by the conjugate of the denominator.
9. MEMORISE THIS: The denominator simplifies to a² – b (difference of squares).

STEP-BY-STEP METHOD

Simplifying Surds

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

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

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

Step 4: Check if the remaining surd can be simplified further. - √2 is already in simplest form.

Rationalising the Denominator (Single Surd)

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

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

Step 3: Simplify the denominator. - √5 × √5 = 5, so the denominator becomes 5.

Step 4: Write the final simplified fraction. - 3√5 / 5.

Rationalising the Denominator (Binomial Surd)

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

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

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

Step 4: Expand the numerator. - Numerator: 6 – 2√2.

Step 5: Write the final simplified fraction. - (6 – 2√2) / 7.

WORKED EXAMPLES

Example 1 – Basic: Simplify √72

Step 1: Factorise 72. - 72 = 36 × 2 (36 is a perfect square).

Step 2: Split the surd. - √72 = √36 × √2.

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

What we did and why: We split 72 into a perfect square (36) and another number (2) to simplify the surd. This makes the expression cleaner and easier to work with in equations.

Example 2 – Medium: Rationalise 5 / √3

Step 1: Multiply numerator and denominator by √3. - (5 × √3) / (√3 × √3).

Step 2: Simplify the denominator. - √3 × √3 = 3.

Step 3: Write the final answer. - 5√3 / 3.

What we did and why: We removed the surd from the denominator by multiplying by √3, which is the standard method for single-surd denominators. This is required for full marks in exams.

Example 3 – Exam-Style: Simplify (4 + √7) / (2 – √7)

Step 1: Identify the conjugate of the denominator. - Denominator: 2 – √7. Conjugate: 2 + √7.

Step 2: Multiply numerator and denominator by the conjugate. - [(4 + √7)(2 + √7)] / [(2 – √7)(2 + √7)].

Step 3: Expand the denominator using difference of squares. - Denominator: 2² – (√7)² = 4 – 7 = –3.

Step 4: Expand the numerator using FOIL (First, Outer, Inner, Last). - (4 × 2) + (4 × √7) + (√7 × 2) + (√7 × √7) = 8 + 4√7 + 2√7 + 7 = 15 + 6√7.

Step 5: Write the final fraction. - (15 + 6√7) / –3.

Step 6: Simplify the fraction. - Divide numerator and denominator by –3: –5 – 2√7.

What we did and why: We used the conjugate to rationalise the denominator, then expanded carefully. The negative denominator was simplified at the end to match exam expectations.

COMMON MISTAKES

1. MISTAKE: Forgetting to simplify surds fully.
2. WHY IT HAPPENS: Students stop at √50 = √25 × √2 but forget to simplify √25 to 5.

3. CORRECT APPROACH: Always check if the surd can be simplified further.

4. MISTAKE: Multiplying by the wrong conjugate.

5. WHY IT HAPPENS: Students use a + √b instead of a – √b (or vice versa).

6. CORRECT APPROACH: The conjugate must have the opposite sign between the terms.

7. MISTAKE: Incorrectly expanding the denominator.

8. WHY IT HAPPENS: Students write (a + √b)(a – √b) = a² + b instead of a² – b.

9. CORRECT APPROACH: Use the difference of squares formula: (a + b)(a – b) = a² – b².

10. MISTAKE: Leaving a negative denominator.

11. WHY IT HAPPENS: Students forget to simplify fractions like (–15 – 6√7) / 3.

12. CORRECT APPROACH: Divide numerator and denominator by the same number to make the denominator positive.

13. MISTAKE: Not rationalising denominators in final answers.

14. WHY IT HAPPENS: Students think 1/√2 is acceptable in exams.
15. CORRECT APPROACH: Always rationalise denominators unless the question specifies otherwise.

EXAM TRAPS

1. TRAP: Disguised surds in physics/chemistry questions.
2. HOW TO SPOT IT: Questions about wave speed (v = fλ), half-life (N = N₀e^(–λt)), or resistors (R = √(R₁² + R₂²)) often hide surds.

3. HOW TO AVOID IT: Look for square roots in formulas. Simplify surds before plugging numbers in.

4. TRAP: Mixed surds in denominators (e.g., 3 / (√2 + 1)).

5. HOW TO SPOT IT: The denominator has a surd and a whole number.

6. HOW TO AVOID IT: Always use the conjugate for binomial denominators.

7. TRAP: "Simplify fully" instructions.

8. HOW TO SPOT IT: The question says "simplify fully" or "give your answer in simplest form."
9. HOW TO AVOID IT: Check for:
   • Unsimplified surds (e.g., √8 instead of 2√2).
   • Unrationalised denominators.
   • Negative denominators.

1-MINUTE RECAP

"Right, listen up—this is your last-minute surds survival guide. First, simplify surds by splitting them into a perfect square and another number. For example, √50 = 5√2. Second, rationalise denominators by multiplying by the surd (for single surds) or the conjugate (for binomials). Remember: the conjugate flips the sign between the terms. Third, expand carefully—FOIL for numerators, difference of squares for denominators. Fourth, always check your final answer for unsimplified surds or negative denominators. And finally, if the question says ‘simplify fully,’ you’ve probably missed a step. Now go ace that exam!"