Revision notes, worked examples and methods for surds and exact calculations.
Notes and quizzes ready · 50 questions · Video coming soon.
This topic is Higher-only.
Revise the key ideas
Higher — exact roots and surds
A surd is an irrational root left in exact form, such as √2. √9 = 3 is rational and is not a surd. An irrational decimal neither terminates nor repeats.
For non-negative a and b, √(ab) = √a × √b. Remove square factors: √72 = √(36 × 2) = 6√2. The sum rule is different: √(a + b) is generally not √a + √b.
Like surds combine like algebraic terms: 3√5 + 2√5 = 5√5, but √2 + √3 cannot be collected into one root.
Worked example: √48 + √27 = 4√3 + 3√3 = 7√3. Simplify the roots before looking for like terms.
Multiplying surds can give a rational result: √3 × √12 = √36 = 6. Keep exact values until the final rounding, if rounding is requested.
Conjugates have opposite signs: (a + √b)(a − √b) = a2 − b. The middle surd terms cancel.
Rationalising removes a surd from a denominator by multiplying numerator and denominator by the same suitable expression. This leaves the fraction's value unchanged.
Worked example:5√2 = 5√22. Multiplying only the denominator would change the value.
Worked example:32 + √3 = 3(2 − √3)(2 + √3)(2 − √3) = 6 − 3√3, since the denominator becomes 4 − 3 = 1.
If a question asks for an exact answer, use a simplified surd, fraction or multiple of π as appropriate. 1.414 is an approximation to √2, not the same exact number.
For a square with area 18 cm2, side length is √18 = 3√2 cm and perimeter is 12√2 cm. Area and length have different units.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
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