Laws of Indices

The Six Laws of Indices

Law	Rule	Example
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ	x³ × x⁵ = x⁸
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	y⁶ ÷ y² = y⁴
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(x²)³ = x⁶
Power of a product	(ab)ⁿ = aⁿbⁿ	(2x)³ = 8x³
Zero index	a⁰ = 1	5⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	x⁻² = 1/x²

Negative Indices

A negative index means take the reciprocal:

• x⁻³ = 1/x³
• 2⁻¹ = 1/2

Fractional Indices

A fractional index represents a root:

• a^(1/n) = ⁿ√a (the nth root of a)
• a^(m/n) = (ⁿ√a)ᵐ

Examples:

• 27^(1/3) = ∛27 = 3
• 16^(3/4) = (⁴√16)³ = 2³ = 8

Mixed Example

Simplify (2x²)³:

• Apply power of a product: 2³ × (x²)³
• = 8 × x⁶ = 8x⁶

Why These Laws Work

The multiplication law arises because: x³ × x⁵ = (x·x·x) × (x·x·x·x·x) = x⁸

Counting the factors gives the sum of exponents.