Multiplying and expanding surds

• √a × √b = √(ab): √6 × √10 = √60 = 2√15.
• √a × √a = a: √7 × √7 = 7.
• Numbers in front multiply separately: 3√2 × 5√6 = 15√12 = 30√3.
• √a ÷ √b = √(a ÷ b): √72 ÷ √2 = √36 = 6.

Expand as with letters, then use √k × √k = k.

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

A square: (4 + √3)² = 16 + 4√3 + 4√3 + 3 = 19 + 8√3. Do not forget the two middle terms.

When the signs are opposite, the surds cancel: (6 + √5)(6 − √5) = 36 − 5 = 31.

Use the calculator to work out (3 + √5)(2 − √5) as it is written, then work out 1 − √5. The two decimals should match.