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).
Diese Lernzettel sind auf Englisch, weil sie britischen Prüfungslehrplänen folgen.
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses