Fractional and negative indices
A negative power takes a reciprocal; a fractional power describes a root.
For nonzero a, a⁻ⁿ = 1/aⁿ. For positive a, a^(m/n) = (ⁿ√a)^m. Keep the conditions alongside the calculation: even roots of negative real numbers do not exist, while odd roots can be negative.
When multiplying equal bases, add exponents. When raising a power to another power, multiply exponents. These rules apply to products and powers, not to sums. For example (x + y)² must be expanded; it is not x² + y².
Worked example
Evaluate 16^(−3/4).
- 16^(1/4) = 2.
- 16^(3/4) = 2³ = 8.
- The negative exponent takes the reciprocal.
Answer: 1/8
Practise fractional and negative indices
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 2.18: Fractional and negative indices
- 7M20 · core · N1.1: Fractional and negative indices
- 8365 · core · 2.18: A fractional or negative index equation
- 7M20 · core · N1.1: A fractional or negative index equation
Next practice: Surds and exact answers, Expanding and factorising.