Laws of Indices

What are Indices?

An index (plural: indices) is the small raised number that shows how many times a base is multiplied by itself. The terms index, exponent, and power all mean the same thing.

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

• Base: 2
• Index: 5

The laws of indices let you simplify expressions involving powers without expanding them fully.

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, x⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	x⁻² = 1/x²

These laws only apply when the base is the same. You cannot combine x³ × y² using the multiplication law because x and y are different bases.

Why the Laws Work

The multiplication law makes sense by counting factors:

x³ × x⁵ = (x · x · x) × (x · x · x · x · x) = x⁸

Counting gives 3 + 5 = 8 total factors.

Similarly, the division law works by cancelling:

x⁵ ÷ x² = (x · x · x · x · x) / (x · x) = x³

Five factors minus two cancelled gives 5 - 2 = 3 remaining.

Negative Indices

A negative index means take the reciprocal (flip it):

a⁻ⁿ = 1/aⁿ

• x⁻³ = 1/x³
• 2⁻¹ = 1/2
• 3⁻² = 1/9

To evaluate a negative index on a number: first find the positive power, then take the reciprocal.

4⁻² = 1/4² = 1/16

Fractional Indices

A fractional index represents a root:

• a^(1/n) = ⁿ√a (the nth root of a)
• a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ) — root first, then power (or vice versa)

Examples:

• 25^(1/2) = √25 = 5
• 27^(1/3) = ∛27 = 3
• 16^(3/4) = (⁴√16)³ = 2³ = 8
• 8^(2/3) = (∛8)² = 2² = 4

Tip: for fractional indices, taking the root first keeps numbers smaller and easier to handle.

Worked Examples

Example 1: Simplify x³ × x⁵

• Same base, add indices: x³⁺⁵ = x⁸

Example 2: Simplify y⁶ ÷ y²

• Same base, subtract indices: y⁶⁻² = y⁴

Example 3: Simplify (2x²)³

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

Example 4: Simplify (a³)⁴ ÷ a⁵

• First: (a³)⁴ = a¹²
• Then: a¹² ÷ a⁵ = a¹²⁻⁵ = a⁷

Example 5: Simplify 3x² × 4x³

• Multiply coefficients: 3 × 4 = 12
• Add indices: x²⁺³ = x⁵
• Answer: 12x⁵

Expressions with Coefficients

When the expression includes a coefficient (number in front), handle the number and the index separately.

(3a²)⁴ = 3⁴ × a² × ⁴ = 81 × a⁸ = 81a⁸

(2x³)² × x = 4x⁶ × x = 4x⁷

Common Mistakes

• Using the multiplication law on different bases: x³ × y⁵ cannot be simplified — different bases.
• Multiplying indices instead of adding when multiplying powers: x³ × x⁵ = x⁸ not x¹⁵.
• Forgetting to apply the power to the coefficient: (2x)³ = 8x³, not 2x³.
• Getting the sign wrong with negative indices: x⁻² = 1/x², not -x².
• Confusing a^(1/2) with a/2: a^(1/2) = √a, which is very different from a ÷ 2.

Tips and Tricks

• Always check the bases are the same before applying a law.
• For fractional indices a^(m/n), use the denominator for the root and the numerator for the power.
• The zero index rule a⁰ = 1 holds for any non-zero base — even 1000000⁰ = 1.
• To simplify complex expressions, break them into steps: handle brackets first, then multiplication/division of indices.
• Negative indices never make the result negative — x⁻² = 1/x² which is positive (assuming x is positive).