An index (or exponent) tells you how many times a base is multiplied by itself. In Form 2 you met positive indices; in Form 3 we extend the idea to zero, negative and fractional indices. The ordinary laws of indices still apply.
Zero and negative indices
Any non-zero number raised to the power 0 equals 1. A negative index means take the reciprocal (turn it upside down).
Key idea
a⁰ = 1 (a ≠ 0)
a⁻ⁿ = 1/aⁿ
a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ
For example 5⁰ = 1 and 2⁻³ = 1/2³ = 1/8. A negative index never makes the answer negative — it produces a fraction.
Fractional indices
A fractional index is a root. The denominator is the root and the numerator is the power.
Worked example
Evaluate 27^(2/3).
Denominator 3 → cube root: ∛27 = 3.
Numerator 2 → square it: 3² = 9.
So 27^(2/3) = (∛27)² = 3² = 9.
Two more: 16^(1/2) = √16 = 4, and 8^(−1/3) = 1/∛8 = 1/2. Always deal with the sign of the index first, then the root, then the power. This ordering avoids mistakes when several rules appear together.
These new indices combine with the old laws. For instance a⁵ ÷ a⁷ = a⁵⁻⁷ = a⁻² = 1/a², which shows why a negative index appears naturally when you divide. Likewise a³ ÷ a³ = a⁰ = 1, confirming the zero-index rule.
Remember
- a⁰ = 1 for any non-zero a.
- A negative index gives a reciprocal, not a negative number.
- In a^(m/n): denominator = root, numerator = power.
- The laws aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ still hold.