Surds

What is a surd?

A surd is an irrational root that cannot be expressed as an exact fraction — it is left in radical form to preserve exactness. For example, √2, √3, √5, 2√3, and 3 + √5 are all surds.

Edexcel 1MA1 Higher papers require exact surd answers in many geometry and algebra questions. Writing a decimal approximation when an exact answer is asked for scores zero.

Simplifying surds

Find the largest perfect-square factor of the number under the root sign.

Rule: √(a × b) = √a × √b

Example: √72 = √(36 × 2) = √36 × √2 = 6√2. Example: √48 = √(16 × 3) = 4√3. Example: √200 = √(100 × 2) = 10√2.

Adding and subtracting surds (like terms only): 3√5 + 7√5 = 10√5. 4√3 − √3 = 3√3. √12 + √27 = 2√3 + 3√3 = 5√3 (simplify first, then combine).

Multiplying surds

√a × √b = √(ab). √a × √a = a (the root of a squared is a).

Example: √3 × √12 = √36 = 6. Example: 2√5 × 3√5 = 6 × 5 = 30.

Expanding brackets with surds

Use FOIL (or the grid method) — exactly as with algebraic expressions.

Example: (3 + √5)(2 − √5) = 6 − 3√5 + 2√5 − (√5)² = 6 − √5 − 5 = 1 − √5.

Example: (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3.

Difference of two squares: (a + √b)(a − √b) = a² − b. This eliminates the surd — used in rationalising.

Rationalising the denominator

Edexcel mark schemes typically require no surd in the denominator.

Simple denominator (√a): Multiply numerator and denominator by √a. Example: 6/√3 = 6√3/3 = 2√3.

Denominator of the form (a + √b) or (a − √b): Multiply by the conjugate (a − √b) or (a + √b). Example: 5/(2 + √3) × (2 − √3)/(2 − √3) = 5(2 − √3)/(4 − 3) = 5(2 − √3)/1 = 10 − 5√3.

Edexcel exam tip

On Paper 1 (non-calculator), surds are the primary way Edexcel tests exact-value arithmetic. Watch for the instruction "give your answer in the form a + b√c" — the answer must be written in that precise form to earn the accuracy mark.