Zero and negative indices

What a⁰ and a⁻ⁿ mean and how to simplify them.

Indices & Logarithms · O/L Maths

Level 1: The zero index

Know why a⁰ = 1

Any number (except 0) raised to the power 0 equals 1: a⁰ = 1.

Why? aᵐ ÷ aᵐ = a^m − m = a⁰. But anything divided by itself is 1.

• (5x)⁰ + 3
• The whole bracket is to the power 0: (5x)⁰ = 1
• 1 + 3 = 4

Level 2: Negative indices

Use a⁻ⁿ = 1 ÷ aⁿ

A negative index means "one over": the reciprocal.

a⁻ⁿ = (1)/(aⁿ)

Keep the halving pattern going: 2² = 4, 2¹ = 2, 2⁰ = 1, 2⁻¹ = ½, 2⁻² = ¼.

• 2⁻³
• Take the reciprocal: 2⁻³ = 1 ÷ 2³
• = (1)/(8)

Level 3: Laws with negative indices

Use the index laws with negative numbers too

The index laws work for negative indices too. With the same base, add (for ×) or subtract (for ÷).

If asked for a positive index in the answer: x⁻³ = 1 ÷ x³.

• x³ ÷ x⁵
• Subtract the indices: 3 − 5 = −2
• x³ ÷ x⁵ = x⁻² = (1)/(x²)

Level 4: Challenge

Equations and reciprocal problems

Write both sides with the same base, then equate the indices. A negative index on the bottom moves to the top and becomes positive.

• ((1)/(2))⁻³
• A negative index on a fraction: flip the fraction.
• ((1)/(2))⁻³ = 2³ = 8