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).