Surds and exact calculation

Surds are irrational square roots left in root form. OCR J560 higher-tier questions regularly ask for answers "in surd form" or "in exact form." Mastering surds prevents unnecessary rounding errors and shows mathematical precision.

What is a surd?

A surd is a root that cannot be simplified to a rational number.

• √4 = 2 → NOT a surd (it simplifies exactly).
• √5, √7, √12, √50 → surds (irrational).

Simplifying surds

Rule: √(a × b) = √a × √b.

Find the largest perfect square factor.

Example: √72 = √(36 × 2) = √36 × √2 = 6√2.

Example: √48 = √(16 × 3) = 4√3.

Example: √200 = √(100 × 2) = 10√2.

Check: 72 = 36 × 2 ✓. Is 36 the largest perfect square factor? Yes.

Adding and subtracting surds

Treat √a as a letter: only collect like surds.

• 3√5 + 4√5 = 7√5 (like terms — both √5).
• 3√2 + 4√3 — cannot be combined (unlike surds).
• √12 + √27 = 2√3 + 3√3 = 5√3 (simplify first, then collect).

Multiplying surds

• √a × √a = a (the definition of a square root).
• √a × √b = √(ab).
• 3√5 × 4√2 = 12√10.

Example: (√3)² = 3. Example: (2√5)² = 4 × 5 = 20.

Rationalising the denominator

The convention is to avoid surds in the denominator. Multiply top and bottom by the surd.

Example: 6/√3 → multiply by √3/√3 → 6√3/3 = 2√3.

Example: 10/(2√5) → multiply by √5/√5 → 10√5/10 = √5.

Rationalising when denominator is (a + √b)

Multiply by the conjugate (a − √b):

Example: 1/(3 + √2) → multiply by (3−√2)/(3−√2) → (3−√2)/(9−2) = (3−√2)/7.

Multiples of π

π is irrational; leave answers involving π in exact form unless told to use π ≈ 3.14159.

Example: circumference of circle radius 5 = 2π × 5 = 10π cm.

Common OCR exam mistakes

1. Trying to add unlike surds: 2√3 + 3√5 ≠ 5√8.
2. Stopping at √72 = √(9 × 8) — wrong! 9 × 8 = 72, but 9 is not the LARGEST perfect square factor (36 is). Always find the largest perfect square factor.
3. (√5)² written as 2√5 — WRONG. (√5)² = 5 (the square and root cancel).
4. Forgetting to rationalise: leaving 3/√7 as the final answer when "exact form" is needed with a rational denominator.