Surds
A surd is a root that does not simplify to a whole number or fraction, such as √2 or √15. Leaving answers as surds keeps them exact, and questions that say "exact value" or "in the form a√b" expect you to.
Surds are on the Higher tier papers for AQA, Edexcel and OCR.
Part 1 of 3: Learn it
In short
- √(ab) = √a × √b, so pull out the largest square factor: √48 = √16 × √3 = 4√3.
- You can only add or subtract like surds: 2√5 + 3√5 = 5√5.
- Rationalise 1/√a by multiplying the top and bottom by √a.
Where this is in your specification
Spec points: DfE N8 (with A4 for expanding brackets); surds are Higher tier
| Board | Topic: Surds |
|---|---|
| DfE content | N8 (with A4, A18 and G20 where surds appear) |
| AQA 8300 | Number |
| Edexcel 1MA1 | Number |
| OCR J560 | 3 Indices and surds |
| Eduqas C300 | HN8 |
| Cambridge IGCSE 0580 | E1.18 |
| National 5 C847 75 | Numerical skills: surds |
The two rules
Also √a × √a = a. There is no rule for √(a + b): √9 + √16 = 3 + 4 = 7, but √25 = 5.
Simplifying
Find the largest square number that is a factor (4, 9, 16, 25, 36, 49, 64, 81, 100) and take its root outside: √72 = √36 × √2 = 6√2; √300 = √100 × √3 = 10√3.
Simplify before you add: √50 + √18 = 5√2 + 3√2 = 8√2.
Expanding brackets
Expand exactly as you would with letters, then tidy up. (3 + √5)(2 − √5) = 6 − 3√5 + 2√5 − 5 = 1 − √5.
Rationalising the denominator
- One surd on the bottom: multiply top and bottom by that surd. 6/√3 = 6√3/3 = 2√3.
- A bracket on the bottom, such as 1/(4 + √2): multiply top and bottom by the same bracket with the sign changed, (4 − √2). The bottom becomes 16 − 2 = 14, a whole number.
Simplify √98.
Show the answer
√49 × √2 = 7√2.
Part 2 of 3: See it worked
Worked examples
Example 1
Simplify √20 + √45.
- √20 = √4 × √5 = 2√5
- √45 = √9 × √5 = 3√5
- 2√5 + 3√5 = 5√5
Answer: 5√5.
Example 2
Rationalise the denominator of 10/(3 − √7). Simplify fully.
- Multiply top and bottom by (3 + √7)
- Bottom: (3 − √7)(3 + √7) = 9 − 7 = 2
- Top: 10(3 + √7) = 30 + 10√7
- (30 + 10√7) ÷ 2
Answer: 15 + 5√7.
Common mistakes
- Writing √(a + b) as √a + √b.
- Taking out a square factor that is not the largest, then stopping: √72 = 2√18 is not fully simplified.
- Adding unlike surds, such as √2 + √3 = √5.
- When rationalising, multiplying only the bottom of the fraction.
Expand and simplify (√6 + 1)².
Show the answer
6 + 2√6 + 1 = 7 + 2√6.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Simplify √98.
√49 × √2 = 7√2.
Expand and simplify (√6 + 1)².
6 + 2√6 + 1 = 7 + 2√6.
Rationalise 15/√5.
15√5/5 = 3√5.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses