Year 8 Maths

Index Laws

Master the rules for working with powers and indices, including the product rule, quotient rule, power of a power, and zero and negative indices.

What are Indices?

An index (or exponent) tells you how many times a number (the base) is multiplied by itself. For example, 2⁴ means 2 × 2 × 2 × 2 = 16.

a m

a = base m = index (exponent)

The Index Laws at a Glance

Law	Rule	Example
Product Rule	a^m × aⁿ = a^m+n	3² × 3⁴ = 3⁶
Quotient Rule	a^m ÷ aⁿ = a^m−n	5⁶ ÷ 5² = 5⁴
Power of a Power	(a^m)ⁿ = a^mn	(2³)⁴ = 2¹²
Zero Index	a⁰ = 1	7⁰ = 1
Negative Index	a^−m = 1/a^m	2⁻³ = 1/8

Product and Quotient Rules

Product Rule — Add the Indices

When multiplying powers with the same base, add the indices.

x³ × x⁵ = x³⁺⁵ = x⁸

2⁴ × 2³ = 2⁷ = 128

Quotient Rule — Subtract the Indices

When dividing powers with the same base, subtract the indices.

x⁷ ÷ x³ = x⁷⁻³ = x⁴

5⁶ ÷ 5⁴ = 5² = 25

Important: These rules only work when the bases are the same!

x³ × y³ cannot be simplified using the product rule because the bases (x and y) are different.

Zero and Negative Indices

Zero Index: a⁰ = 1

Any non-zero number raised to the power of zero equals 1. This follows from the quotient rule:

a³ ÷ a³ = a⁰ = 1 (since any number ÷ itself = 1)

100⁰ = 1

(xy)⁰ = 1

Negative Index: a^−m = 1/a^m

A negative index means the reciprocal of the positive power.

2⁻¹ = 1/2 = 0.5

3⁻² = 1/3² = 1/9 = 0.111...

x⁻⁴ = 1/x⁴

Key Vocabulary

Index (Exponent)

The small number written above and to the right of the base, indicating how many times the base is multiplied by itself.

Base

The number or variable that is being raised to a power. In 5³, the base is 5.

Reciprocal

The multiplicative inverse of a number. The reciprocal of a^m is 1/a^m, which equals a^−m.

Power

The result of raising a base to an index. For example, 2³ = 8, so 8 is the third power of 2.

Worked Examples

Simplify: x⁴ × x⁶ ÷ x³

Step 1: Apply the product rule: x⁴ × x⁶ = x⁴⁺⁶ = x¹⁰

Step 2: Apply the quotient rule: x¹⁰ ÷ x³ = x¹⁰⁻³ = x⁷

Answer: x⁷

Evaluate: (2³)²

Step 1: Apply the power of a power rule: (2³)² = 2^3×2 = 2⁶

Step 2: Evaluate: 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64

Answer: 64

Write 4⁻² as a fraction and as a decimal.

Step 1: Apply the negative index rule: 4⁻² = 1/4²

Step 2: Evaluate: 1/4² = 1/16 = 0.0625

Answer: 1/16 or 0.0625

Knowledge Check

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Key Concepts Summary

●Product Rule: a^m × aⁿ = a^m+n — add indices when multiplying same base.
●Quotient Rule: a^m ÷ aⁿ = a^m−n — subtract indices when dividing same base.
●Power of a Power: (a^m)ⁿ = a^mn — multiply the indices.
●Zero Index: a⁰ = 1 for any non-zero a.
●Negative Index: a^−m = 1/a^m — flip to the denominator.

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