Indices and exponent rules: the laws of indices explained
Learn the index laws for multiplying, dividing and raising powers, plus zero and negative indices. UK indices and US exponent rules, with examples.
GCSEHigh SchoolCurriculum fit: GCSE Maths Number and Algebra (Foundation and Higher; fractional indices are Higher). US: Algebra I exponent rules, Algebra II rational exponents.
On this page
What is it?
An index (also called a power or exponent) tells you how many times to multiply a number by itself. In 2⁵ the base is 2 and the index is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
Index laws are shortcuts for combining powers of the same base. They work for numbers and for letters in algebra.
Key formulas and ideas
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1
- a⁻ⁿ = 1/aⁿ
- a^(1/2) = √a (Higher)
Worked examples
Example 1: Multiplying powers
Question: Simplify 3x⁴ × 2x⁵.
- Multiply the numbers: 3 × 2 = 6.
- Add the indices of x: 4 + 5 = 9.
Answer: 6x⁹
Example 2: Dividing and powers of powers
Question: Simplify (2x³)² ÷ x².
- Square everything inside the bracket: 2² × x⁶ = 4x⁶.
- Divide: x⁶ ÷ x² = x⁴.
Answer: 4x⁴
Example 3: Zero and negative indices
Question: Work out 5⁰ and 2⁻³.
- Anything to the power 0 is 1, so 5⁰ = 1.
- 2⁻³ = 1/2³ = 1/8.
Answer: 1 and 1/8
Common mistakes
- Multiplying the indices when you should add them, such as x² × x³ = x⁶ (it is x⁵).
- Applying the laws to different bases, such as 2³ × 3² = 6⁵ (it is not).
- Thinking a negative index makes the answer negative: 2⁻³ is 1/8, not −8.
- Forgetting to apply a power to the number in front: (2x)³ = 8x³, not 2x³.
Practice questions
Frequently asked questions
Why is a⁰ equal to 1?
Using the division law, aⁿ ÷ aⁿ = a⁰. But any number divided by itself is 1, so a⁰ = 1.
Do the laws work when the bases are different?
No. The multiplication and division laws need the same base.
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Reviewed by StudyMorrow editorial review on 5 Nov 2026.