M6 · Surds and exact calculations | Edexcel GCSE Maths

Revise the key ideas

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.

Expand surd brackets term by term: (√3 + 2)(√3 − 1) = 3 − √3 + 2√3 − 2 = 1 + √3.
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.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.