Laws of indices
Multiply, divide and raise powers: aᵐ × aⁿ, aᵐ ÷ aⁿ, (aᵐ)ⁿ.
Indices & Logarithms · O/L Maths
Level 1: What is a power?
Read and evaluate powers like 2³
A power is a short way to write repeated multiplication. 2⁴ means 2 × 2 × 2 × 2.
The big number is the base. The small raised number is the index (also called the exponent). It tells you how many times to multiply the base.
- 3⁴
- Expand: 3⁴ = 3 × 3 × 3 × 3
- 3 × 3 = 9
- 9 × 3 = 27
- 27 × 3 = 81, so 3⁴ = 81
Level 2: Multiplying powers
Use aᵐ × aⁿ = a^m + n
When you multiply powers with the same base, add the indices.
aᵐ × aⁿ = a^m + n
Why? a³ × a² = (a × a × a) × (a × a) = a⁵. Try it in the explorer below.
- 2³ × 2⁴
- The base is the same (2). Add the indices.
- 2³ × 2⁴ = 2^3 + 4 = 2⁷
- 2⁷ = 128
Level 3: Dividing powers
Use aᵐ ÷ aⁿ = a^m − n
When you divide powers with the same base, subtract the indices.
aᵐ ÷ aⁿ = a^m − n
Why? In a⁵ ÷ a², the two a’s on the bottom cancel two of the a’s on top, leaving a³.
- 5⁶ ÷ 5²
- The base is the same (5). Subtract the indices.
- 5⁶ ÷ 5² = 5^6 − 2 = 5⁴
- 5⁴ = 625
Level 4: Power of a power
Use (aᵐ)ⁿ = a^m × n
When you raise a power to another power, multiply the indices.
(aᵐ)ⁿ = a^m × n
Raise every factor inside the brackets to the power: (2x³)² = 2² × x⁶ = 4x⁶.
- (2³)²
- (2³)² means 2³ × 2³.
- Multiply the indices: 3 × 2 = 6
- (2³)² = 2⁶ = 64
Level 5: Combining the laws
Simplify using several laws in one problem
Work step by step: brackets first (power of a power), then multiplying, then dividing. Use one law at a time.
- (x³ × x⁴) ÷ x²
- Multiply first: add the indices. x³ × x⁴ = x⁷
- Then divide: subtract the indices. x⁷ ÷ x² = x⁵
- (x³ × x⁴) ÷ x² = x⁵
Level 6: Challenge
Exam-style problems: solving for x
If two powers with the same base are equal, their indices must be equal.
Sometimes you must first write a number as a power of a smaller base: 4 = 2², 8 = 2³, 9 = 3², 27 = 3³.
- 4ˣ = 2⁶
- Write 4 as 2².
- (2²)ˣ = 2²ˣ = 2⁶
- Equate the indices: 2x = 6, so x = 3