Surds & Exact Calculations
Surds & Exact Calculations (Higher)
A surd is a root that cannot be simplified to a whole number or exact fraction — for example √2, √3 and √5. Surds give exact answers rather than rounded decimals.
Why Use Surds?
√2 = 1.41421356... This decimal never terminates or repeats. Writing √2 is perfectly exact; writing 1.414 is an approximation. Exam questions that say "give an exact answer" or "leave in surd form" require surds.
Simplifying Surds
Use the rule: √(a × b) = √a × √b
Look for the largest square number that is a factor.
Worked Example: Simplify √72.
- 72 = 36 × 2
- √72 = √36 × √2 = 6√2
Worked Example: Simplify √200.
- 200 = 100 × 2
- √200 = √100 × √2 = 10√2
Common square factors to look for: 4, 9, 16, 25, 36, 49, 64, 100.
Rules for Surds
| Rule | Example |
|---|---|
| √a × √b = √(ab) | √3 × √5 = √15 |
| √a ÷ √b = √(a/b) | √20 ÷ √5 = √4 = 2 |
| (√a)² = a | (√7)² = 7 |
| a√n + b√n = (a+b)√n | 3√2 + 5√2 = 8√2 |
| a√n − b√n = (a−b)√n | 7√3 − 2√3 = 5√3 |
You can only add or subtract surds with the same root.
Worked Example: Simplify √12 + √27.
- √12 = √(4×3) = 2√3
- √27 = √(9×3) = 3√3
- 2√3 + 3√3 = 5√3
Expanding Brackets with Surds
Apply the same expansion rules as algebra.
Worked Example: Expand and simplify (2 + √3)(4 − √3).
- = 2×4 + 2×(−√3) + √3×4 + √3×(−√3)
- = 8 − 2√3 + 4√3 − 3
- = 5 + 2√3
Worked Example: Expand (3 + √5)².
- = (3 + √5)(3 + √5)
- = 9 + 3√5 + 3√5 + 5
- = 14 + 6√5
Rationalising the Denominator
A surd in the denominator is considered unsimplified. To rationalise, remove the surd from the bottom.
Simple case: Multiply top and bottom by the surd.
Worked Example: Rationalise 6/√3.
- = (6 × √3) / (√3 × √3)
- = 6√3 / 3
- = 2√3
Two-term denominator: Multiply by the conjugate (change the sign between the terms).
Worked Example: Rationalise 5/(3 + √2).
- Multiply by (3 − √2)/(3 − √2)
- Numerator: 5(3 − √2) = 15 − 5√2
- Denominator: (3 + √2)(3 − √2) = 9 − 2 = 7
- = (15 − 5√2) / 7
The denominator simplifies because (a + b)(a − b) = a² − b², and the surd squares away.
Exam Tips
- Always simplify surds fully — look for square factors you might have missed
- "Give an exact answer" means leave as a surd or fraction — do not use a calculator to get a decimal
- When rationalising a two-term denominator, remember to multiply BOTH top and bottom by the conjugate