A surd is an irrational root, such as √2 or √7, left in exact form. Simplifying a surd means pulling out any perfect square, and rationalising means rewriting the denominator so it contains no surd.
This lesson is part of SPM Additional Mathematics indices, surds and logarithms. If roots as powers still feel unsteady, read applying index laws with fractional powers first.
How do I simplify a surd?
Find the largest perfect square that divides the number. Write the number as that square times the remainder, then take the square root of the square.
√72 = √(36 × 2) = √36 × √2 = 6√2.
Surds can be added and subtracted only when the surd part is the same. Simplify each first:
√50 + √8 − √18 = 5√2 + 2√2 − 3√2 = 4√2.
Worked example: rationalising a denominator
Write 6 ÷ √3 without a surd in the denominator. Multiply top and bottom by √3, since √3 × √3 = 3:
6 ÷ √3 = (6 × √3) ÷ (√3 × √3) = 6√3 ÷ 3 = 2√3.
When the denominator has two terms, use the conjugate. Rationalise 4 ÷ (3 − √5). The conjugate of 3 − √5 is 3 + √5.
Denominator: (3 − √5)(3 + √5) = 9 − 5 = 4.
Numerator: 4(3 + √5).
So 4(3 + √5) ÷ 4 = 3 + √5.
The step that does the work is the difference of two squares, (a − b)(a + b) = a² − b². It makes the cross terms cancel.
The mistake that costs marks
The common slip is to expand (√3 + √2)² as 3 + 2 = 5. The middle term is missing, so the result looks tidy and is wrong.
| Step | Wrong | Right |
|---|---|---|
| Expand | (√3)² + (√2)² | (√3)² + 2(√3)(√2) + (√2)² |
| Simplify | 3 + 2 | 3 + 2√6 + 2 |
| Answer | 5 | 5 + 2√6 |
A number check exposes it. √3 + √2 ≈ 3.146, and squaring gives about 9.9. The right-hand column gives 5 + 2√6 ≈ 9.9, while 5 is nowhere near.
A method you can reuse
- Simplify every surd first, pulling out perfect squares.
- Group like surds and combine them.
- For a single-term denominator √a, multiply top and bottom by √a.
- For a two-term denominator, multiply top and bottom by its conjugate.
- Finish by simplifying the numerator and cancelling any common factor.
Record which step slips in the mistake log, so your practice targets it.
Check yourself
Rationalise 6 ÷ (√5 + 1) and write the answer in the form (a√5 − b) ÷ c.
Answer
The conjugate of √5 + 1 is √5 − 1.
Denominator: (√5 + 1)(√5 − 1) = 5 − 1 = 4.
Numerator: 6(√5 − 1) = 6√5 − 6.
So the fraction is (6√5 − 6) ÷ 4, which simplifies to (3√5 − 3) ÷ 2.
Check with decimals: 6 ÷ (2.236 + 1) = 1.854, and (3 × 2.236 − 3) ÷ 2 = 1.854.
What to study next
Logarithms are the next inverse tool. Continue with using logarithm laws with domain checks, then try the chapter practice set.
If you want a teacher to work through exact-form questions with you, see online one-to-one Additional Mathematics tuition.