Surds

A surd is a root that doesn’t simplify to a whole number, like √2 or √12. Keeping answers as surds keeps them exact. We simplify surds by taking out square factors, and rationalise denominators so there’s no root on the bottom.

Key facts

  • √(ab) = √a × √b
  • √a × √a = a
  • Simplify by finding the biggest square factor: √50 = √25 × √2 = 5√2.
  • Rationalise a/√b by multiplying top and bottom by √b.
  • For 1/(a + √b) multiply by (a − √b).

Worked examples

Example 1

Simplify √72.

  1. 72 = 36 × 2, and 36 is square.
  2. √72 = √36 × √2 = 6√2

Answer: 6√2

Example 2

Rationalise 6/√3.

  1. 6/√3 × √3/√3 = 6√3/3
  2. Simplify. = 2√3

Answer: 2√3

Example 3

Expand (3 + √2)(3 − √2).

  1. (a + b)(a − b) = a² − b²
  2. 9 − 2 = 7

Answer: 7