Powers and Indices: Laws, Formulas and Examples

A power is shorthand for repeated multiplication of a number by itself. It is written aⁿ, where a is the base and n is the index or exponent. For example 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The topic is taught in Year 9 and developed further in Year 10.

The Eight Index Laws

Why a⁰ = 1

This is the question children ask most, and answering "that is just the rule" is exactly why they forget it by tomorrow.

Look at the division pattern: 2⁵ ÷ 2³ = 2². Now take 2³ ÷ 2³. By law two the answer is 2⁰. But any number divided by itself is 1. So 2⁰ must be 1.

Another route: step the index down, halving each time → 2³ = 8, 2² = 4, 2¹ = 2, so 2⁰ = 1. The pattern holds.

Why a Negative Index Means a Fraction

Continue the same pattern downwards: 2⁰ = 1, then 2⁻¹ = 1/2, 2⁻² = 1/4. A negative index does not make the answer negative — that is the number one misconception in this topic. 2⁻³ = 1/8, not −8.

Standard Form

Powers let very large or very small numbers be written compactly:

• 5,700,000 = 5.7 × 10⁶
• 0.00042 = 4.2 × 10⁻⁴

The rule: the leading number must be between 1 and 10. This form is used across science, and in computing too — storage works in powers of two, so 1 KB is really 2¹⁰ = 1,024 bytes, not 1,000.

Worked Examples

Simplify: (3² × 3⁴) ÷ 3³

• Numerator: 3²⁺⁴ = 3⁶
• Divide: 3⁶⁻³ = 3³ = 27

Evaluate: (2⁻²)⁻³

• Multiply the indices: 2^((−2)×(−3)) = 2⁶ = 64

The Mistakes That Actually Cost Marks

1. Thinking a negative index gives a negative answer. 2⁻³ is 1/8, not −8. The number one cause, and it usually survives into senior school if not corrected.

2. Adding the bases instead of the indices. Writing 2³ × 2⁴ = 4⁷. The base stays 2; only the indices add.

3. Applying the laws across different bases. 2³ × 3² cannot be collapsed into a single power. Laws 1 and 2 require the same base.

4. Misplacing the minus sign. −2⁴ = −16, but (−2)⁴ = 16. The brackets change the whole answer, and this is a favourite trap question.

5. Thinking a⁰ = 0. It is 1, and the reason can be derived rather than memorised, as shown above.