Overview
Manipulating powers (indices) and roots (surds) accurately and exactly.
Laws of indices
- aᵐ × aⁿ = a^(m+n)
- aᵐ ÷ aⁿ = a^(m−n)
- (aᵐ)ⁿ = a^(mn)
- a⁰ = 1
- a⁻ⁿ = 1 ÷ aⁿ
- a^(1/n) = ⁿ√a, so a^(m/n) = (ⁿ√a)ᵐ
Surds
A surd is an irrational root left in exact form (e.g. √2).
- √(ab) = √a × √b → simplify, e.g. √12 = √4 × √3 = 2√3.
- Rationalising: remove a surd from the denominator by multiplying top and
bottom by it (or by the conjugate).
Worked examples
- Simplify 16^(3/4) = (⁴√16)³ = 2³ = 8.
- Rationalise 1 ÷ √3 = (1 × √3) ÷ (√3 × √3) = √3 ÷ 3.
Common mistakes
- Writing √a + √b = √(a + b) — this is not true.
Exam tips
- "Give an exact answer" usually means leave it as a surd, don't round.