Surds
Section: Number 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
The Laws of Surds
Extended Only
A surd is a root that cannot be simplified to a whole number, such as √3 or √7. Multiplying and dividing surds each follow a simple law, but addition and subtraction work completely differently - and knowing why matters just as much as knowing the rule.
- Multiplication law: √(a) × √(b) = √(ab)
- Division law: √(a) ÷ √(b) = √(a/b)
- Squaring law: (√(a))^2 = a - squaring undoes a square root
- Addition/subtraction law: a√(n) ± b√(n) = (a ± b)√(n) - but this only works when the number under the root, n, is identical in both surds
- Unlike multiplication and division, there is no law that combines √a + √b into a single surd when a ≠ b
Worked Example: Applying the Laws of Surds
- Multiplication: √(3) × √(12) = √(36) = 6
- Division: √(50) ÷ √(2) = √(25) = 5
- Squaring: (√(11))^2 = 11
- Addition (like surds only): 4√(6) + 3√(6) = 7√(6)
Common Mistakes
MistakeAssuming √a + √b = √(a+b) by analogy with the multiplication law √a × √b = √(ab)
Fixcheck with numbers: √9 + √16 = 3 + 4 = 7, but √(9+16) = √25 = 5 - these are different, so no such addition law exists
MistakeTrying to combine unlike surds into a single surd, e.g. writing 2√3 + 5√7 as some single root
Fixsurds with different numbers under the root simply cannot be combined into one term - 2√3 + 5√7 is already fully simplified as it stands
Simplifying Surds
Extended Only
The multiplication law also works in reverse: splitting the number under a root into a perfect square factor and another factor is the key technique for simplifying a surd, and for spotting when two surds can be added or subtracted.
- To simplify a surd, find the largest perfect square factor of the number under the root, and take its square root outside, using the multiplication law in reverse: √(ab) = √(a) × √(b)
- Two surds that look "unlike" may become like surds once each is simplified - always simplify first before deciding whether they can combine
Worked Example: Simplifying a Surd
- Question: Simplify √75.
Give your answer in the form a√b.- Step 1: The largest perfect square factor of 75 is 25: 75 = 25 × 3
- Step 2: Split the root: √(75) = √(25) × √(3) = 5√(3)
- Answer: 5√3
Worked Example: Adding and Subtracting Surds
- Question: Simplify.
(i) √18 + √98
(ii) √48 - √12- Step 1 (i): √(18) = √(9 × 2) = 3√(2), √(98) = √(49 × 2) = 7√(2)
- Step 2 (i): Both are now multiples of √2, so combine using the addition law: 3√(2) + 7√(2) = 10√(2)
- Step 1 (ii): √(48) = √(16 × 3) = 4√(3), √(12) = √(4 × 3) = 2√(3)
- Step 2 (ii): 4√(3) - 2√(3) = 2√(3)
- Answer: (i) 10√2, (ii) 2√3
Common Mistakes
MistakeNot using the largest perfect square factor, leaving a surd not fully simplified, e.g. writing √48 = √4 × √12 = 2√12
Fix2√12 is not fully simplified, since √12 itself simplifies further to 2√3 - always check the perfect square factor used is the largest one: √48 = √16 × √3 = 4√3
MistakeConcluding two surds cannot be added just because they look different, without simplifying first, e.g. leaving √8 + √18 as "cannot be combined"
Fixsimplify each surd first: √8 = 2√2 and √18 = 3√2 - once simplified, these are like surds and combine to 5√2
Rationalising the Denominator
Extended Only
A fraction with a surd in the denominator can be rewritten with a whole number denominator instead. Multiplying top and bottom by the right expression removes the surd from the bottom without changing the fraction's value.
- For a denominator that is just a surd, multiply top and bottom by that same surd: (a/√(b)) = (a/√(b)) × (√(b)/√(b)) = (a√(b)/b)
- For a denominator with two terms (one involving a surd), multiply top and bottom by the conjugate - the same two terms with the sign between them reversed
- Multiplying a sum by its conjugate removes the surd, since (x+y)(x-y) = x^2 - y^2
Worked Example: Rationalising a Simple Denominator
- Question: Simplify (18/√(6)).
Give your answer in the form a√b.- Step 1: Multiply top and bottom by √6: (18/√(6)) × (√(6)/√(6)) = (18√(6)/6)
- Step 2: Simplify: (18√(6)/6) = 3√(6)
- Answer: 3√6
Worked Example: Rationalising a Denominator with the Conjugate
- Question: Simplify (1/2+√(5)).
Give your answer in the form a + b√c.- Step 1: Multiply top and bottom by the conjugate, 2 − √5: (1/2+√(5)) × (2-√(5)/2-√(5))
- Step 2: Expand the denominator: (2+√(5))(2-√(5)) = 2^2 - (√(5))^2 = 4 - 5 = -1
- Step 3: The fraction becomes (2-√(5)/-1) = √(5) - 2
- Answer: √5 − 2
Common Mistakes
MistakeMultiplying only the denominator by the surd or conjugate, and leaving the numerator unchanged
Fixwhatever is multiplied onto the denominator must also be multiplied onto the numerator, so the fraction's value does not change
MistakeUsing the same sign for the conjugate instead of reversing it, e.g. multiplying (2+√5) by (2+√5) again
Fixthe conjugate must have the opposite sign in the middle - multiplying by the same expression again would not remove the surd from the denominator
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