Multiplying and expanding surds
Multiplying and dividing
- √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.
Expanding brackets
Expand as with letters, then use √k × √k = k.
- (3 + √5)(2 − √5)
- = 6 − 3√5 + 2√5 − 5
- = 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.
Expand, then check
Use the calculator to work out (3 + √5)(2 − √5) as it is written, then work out 1 − √5. The two decimals should match.
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