Deriving the zero index
Any non-zero number divided by itself equals 1. Using the quotient law:
7⁴ ÷ 7⁴ = 1 and 7⁴⁻⁴ = 7⁰
Therefore 7⁰ = 1. In general, a⁰ = 1 for a ≠ 0. The expression 0⁰ is not assigned this Stage 4 rule.
Zero Index and Indices Synthesis
Think first: Why does every non-zero number raised to the power zero equal 1?
Open practice worksheet
Know
- the zero-index law for non-zero bases
- which index law applies to each operation
Understand
- how the zero-index law follows from division
- the boundary between Stage 4 positive/zero indices and later negative indices
Can do
- justify a⁰ = 1 for a ≠ 0
- simplify mixed numerical index expressions
- solve and explain unfamiliar index problems
Choose the law from the operation
multiply same base → add
divide same base → subtract
power of power → multiply
zero index non-zero base → 1
Mixed calculations
Apply brackets and indices before remaining operations:
(2³)² ÷ 2⁴ + 5⁰ = 2⁶ ÷ 2⁴ + 1 = 2² + 1 = 5
Negative indices are Stage 5 extension and are deliberately not used here.
Worked examples
Worked example 1: Evaluate 13⁰
13 is non-zero, so 13⁰ = 1.
Worked example 2: Simplify 4⁷ ÷ 4⁷
Subtract indices: 4⁰ = 1, matching the fact that a non-zero number divided by itself is 1.
Worked example 3: Evaluate (3²)³ ÷ 3⁴ + 2⁰
(3²)³ = 3⁶; 3⁶ ÷ 3⁴ = 3² = 9; 2⁰ = 1. Total = 10.
Check your understanding
Reason and apply
1. Use two different arguments to explain why 9⁰ = 1.
Model response: Quotient law: 9³ ÷ 9³ = 9⁰, while any non-zero number divided by itself is 1. Pattern: 9² = 81, 9¹ = 9, and dividing by 9 again gives 9⁰ = 1.
2. Simplify (5²)³ ÷ 5⁴.
Model response: 5⁶ ÷ 5⁴ = 5² = 25.
3. Evaluate 2⁵ × 2² ÷ 2⁶ + 7⁰.
Model response: 2⁷ ÷ 2⁶ + 1 = 2¹ + 1 = 3.
4. Create a numerical expression that uses all four Stage 4 index laws and evaluates to 17.
Model response: Example: (2²)³ ÷ 2⁴ + 3⁰ + 2³ × 2⁰ = 2² + 1 + 8 = 13, not 17; revise to (2²)³ ÷ 2³ + 3⁰ + 2³ × 2⁰ = 8 + 1 + 8 = 17.