Surds
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Surds
Cambridge (CIE) IGCSE Extended Mathematics – Complete Revision Guide
Simplifying Surds
What Is a Surd?
A surd is the square root of a non-perfect square number. The word "surd" refers to an irrational number—a number that cannot be written as a simple fraction and whose decimal expansion goes on forever without repeating. Using surds allows you to express answers exactly, without rounding or approximation.
Surds vs. Non-Surds
These ARE surds: √2, √3, √5, √7, √11 — because 2, 3, 5, 7, and 11 are not perfect squares.
These are NOT surds: √4 = 2, √9 = 3, √16 = 4 — because the results are whole numbers (rational). Also, 2/3 and 6.23 are not surds because they aren't square roots of integers.
Why use surds? The exact value √5 is more accurate and elegant than the decimal approximation 2.236... Using surds preserves precision throughout your working and avoids accumulated rounding errors.
Operations with Surds
Surds follow special rules for multiplication, division, addition, and subtraction. Understanding these rules is essential for working with surds confidently.
Multiplying Surds
You can multiply surds by multiplying the numbers under the square roots together, then simplifying.
√3 × √5 = √(3 × 5) = √15
Dividing Surds
You can divide surds by dividing the numbers under the square roots, then simplifying.
√21 ÷ √7 = √(21 ÷ 7) = √3
Adding and Subtracting Surds
You can only add or subtract surds if they are "like" surds—meaning the same number is under the square root.
3√5 + 8√5 = 11√5 (same surd: √5)
7√3 − 4√3 = 3√3 (same surd: √3)
2√3 + 4√6 — cannot be simplified (different surds)
√9 + √4 = 3 + 2 = 5 (these are not surds; they simplify to whole numbers)
√9 + √4 ≠ √(9 + 4). Never add numbers inside the square root! √9 + √4 = 3 + 2 = 5, but √13 ≈ 3.6.
How Do I Simplify Surds?
To simplify a surd, you extract any perfect square factors from under the square root. This is the most important skill for working with surds.
Find the largest perfect square that divides the number, then separate it out.
Step-by-Step Process
- Find the largest perfect square factor of the number under the root
- Rewrite the number as a product: (perfect square) × (remaining number)
- Split the square root: √(perfect square × remaining) = √(perfect square) × √(remaining)
- Simplify the square root of the perfect square
Step 1: Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Perfect square factors: 1, 4, 16. Largest is 16.
Step 2: 48 = 16 × 3
Step 3: √48 = √(16 × 3) = √16 × √3
Step 4: √16 = 4, so √48 = 4√3
720 = 144 × 5 (since 144 = 12²)
√720 = √(144 × 5) = √144 × √5 = 12√5
Why use the largest perfect square? Using the largest ensures your answer is fully simplified in one step. If you use a smaller perfect square, you'll need to simplify again, which is extra work.
Simplifying Multiple Surds & Collecting Like Terms
When you have multiple surds in an expression, simplify each one separately first. Then collect any like surds together—just like collecting like terms in algebra.
Step 1: Simplify √32: 32 = 16 × 2, so √32 = 4√2
Step 2: Simplify √8: 8 = 4 × 2, so √8 = 2√2
Step 3: Collect like terms: 4√2 + 2√2 = 6√2
Expanding Brackets with Surds
Multiply brackets containing surds just like you would with algebra. Use FOIL (First, Outer, Inner, Last), then simplify using the property (√a)² = a.
FOIL:
First: √6 × √6 = (√6)² = 6
Outer: √6 × 4 = 4√6
Inner: −2 × √6 = −2√6
Last: −2 × 4 = −8
Combine: 6 + 4√6 − 2√6 − 8 = −2 + 2√6
Rationalising Denominators
What Does Rationalising the Denominator Mean?
When a fraction has a surd in the denominator (bottom), we say it has an "irrational denominator." Rationalising means rewriting the fraction so the denominator contains no surds—it becomes rational (a whole number or simple fraction).
Examples of Irrational vs. Rationalised
Irrational denominators: 4/√5, 2/√3
Rationalised form: 4√5/5, 2√3/3
Why rationalise? In pure mathematics, we prefer rational denominators because they're cleaner and easier to work with. It's also the standard form expected in exam answers.
Rationalising Simple Denominators (Just a Surd)
If the denominator is just a single surd (like √5), multiply both the numerator and denominator by that same surd.
Why This Works
When you multiply √b × √b, you get (√b)² = b, which is a whole number. So the denominator becomes rational.
Multiply by √5/√5:
4/√5 × √5/√5 = 4√5/5
Denominator is now 5 (rational) ✓
Rationalising Harder Denominators (Expression with a Surd)
When the denominator is an expression like (3 + √5), you can't just multiply by the surd. Instead, multiply by the expression with the sign changed. This uses the difference of two squares to eliminate the surd.
Why the Difference of Two Squares?
When you multiply (b + √c)(b − √c), the middle terms cancel, leaving b² − c (a rational number).
Step 1: Multiply by (3 − √5)/(3 − √5)
= [2(3 − √5)]/[(3 + √5)(3 − √5)]
Step 2: Expand numerator: 2(3 − √5) = 6 − 2√5
Step 3: Expand denominator using difference of squares:
(3 + √5)(3 − √5) = 3² − (√5)² = 9 − 5 = 4
Step 4: Result: (6 − 2√5)/4 = (3 − √5)/2
Multiply by (√6 + 2)/(√6 + 2):
= [4(√6 + 2)]/[(√6 − 2)(√6 + 2)]
Numerator: 4√6 + 8
Denominator: (√6)² − 2² = 6 − 4 = 2
Result: (4√6 + 8)/2 = 2√6 + 4
If the denominator is (a + √b), multiply by (a − √b). If it's (a − √b), multiply by (a + √b). Always flip the sign!
If your final answer still has a surd on the bottom, you've made a mistake. Go back and check your working!
Summary & Review
The Big Picture
Surds are irrational numbers that let us express exact mathematical values. This chapter teaches two core skills: simplifying surds to their cleanest form, and rationalising denominators so fractions are in standard form.
- What surds are: Square roots of non-perfect squares. They preserve exact values without decimal rounding.
- Operations: Multiply/divide the numbers under the roots. Add/subtract only like surds. Expand brackets like algebra, using (√a)² = a.
- Simplification: Extract perfect square factors. This is the heart of surd manipulation.
- Rationalising: Remove surds from denominators. Simple case: multiply by the surd. Hard case: use difference of two squares.
Key Formulas You Must Know
| Operation | Formula | Example |
|---|---|---|
| Multiply surds | √a × √b = √(ab) | √3 × √5 = √15 |
| Divide surds | √a ÷ √b = √(a/b) | √21 ÷ √7 = √3 |
| Squaring a surd | (√a)² = a | (√5)² = 5 |
| Simplify surds | √(ab) = √a × √b | √48 = √(16 × 3) = 4√3 |
| Like surds | a√b + c√b = (a+c)√b | 3√5 + 8√5 = 11√5 |
| Difference of squares | (a+√b)(a−√b) = a² − b | (3+√5)(3−√5) = 9 − 5 = 4 |
Concepts Checklist
Check off each concept only when you could confidently teach it to someone else.
Simplifying Surds
Rationalising Denominators
What to Memorize
These formulas, patterns, and concepts must be automatic in the exam.
Core Rules (Non-Negotiable)
1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
√(ab) = √a × √b — allows you to split surds for simplification
Only add/subtract surds with the same number under the root (like 3√5 + 8√5)
(a + √b)(a − √b) = a² − b
Common Mistakes to Avoid
Adding or subtracting unlike surds. √3 + √5 cannot be combined. Only like surds can be added.
Adding inside the square root. √9 + √4 ≠ √(9+4). Always simplify first: 3 + 2 = 5 ≠ √13.
Not simplifying fully. √48 ≠ 4√3 until you extract the perfect square factor. Always find the largest perfect square.
Not flipping the sign when rationalising. For 1/(2+√3), multiply by (2−√3), not (2+√3).
Leaving a surd in the denominator. If your final answer has √ on the bottom, you haven't rationalised correctly.
Exam Tips & Strategy
How Surds Are Tested
- Simplification questions: "Simplify √48" or "Write 3√2 + 5√2 in simplest form" — test your ability to extract factors
- Operations: "Work out √18 × √2" or "Divide √24 by √3" — test multiplication and division rules
- Expanding brackets: "Expand (2 + √3)(3 − √3)" — test algebra with surds and (√a)² = a
- Rationalising: "Write 5/(2+√3) in the form a + b√c" — test rationalising and algebraic manipulation
- Combined: "Simplify then rationalise" — tests multiple skills in sequence
Before the Exam
- Memorise perfect squares: Know 1–15 squared instantly. This is the foundation of surd simplification.
- Practice extraction: Find the largest perfect square factor of random numbers. This skill is essential.
- Master the difference of squares: (a+√b)(a−√b) = a² − b. This pattern unlocks hard rationalising.
- Check your final answers: No surds on the bottom? No unlike surds? Fully simplified?
During the Exam
Before doing anything else with a surd expression, simplify each individual surd. This often reveals like surds you can combine.
When simplifying √48, write "48 = 16 × 3" explicitly. This shows your working and prevents errors.
Simple surd in denominator → multiply by that surd. Expression in denominator → multiply by the expression with the sign flipped. Follow the pattern.
When multiplying numerators and denominators, use brackets: [2(3−√5)] and [(3+√5)(3−√5)]. This prevents sign errors.
Before moving to the next question, check: (1) Is every surd simplified? (2) Are like surds collected? (3) If rationalising, is there a surd on the bottom?
Time Management
- Simplification question: 2–3 minutes. If you're taking longer, you haven't memorised perfect squares.
- Rationalising simple denominator: 2–3 minutes. Straightforward process once you know the pattern.
- Rationalising hard denominator: 4–5 minutes. Requires careful algebra and the difference of two squares.
- Multi-step (simplify + rationalise): 5–7 minutes. Break it into smaller steps and don't rush.
Cambridge (CIE) IGCSE Extended Mathematics
Surds — Complete Revision Guide
Tick off the checklist as you master each concept. Surds are a cornerstone of higher-level maths—solidify them now.
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