The laws of indices
An index (or power) tells you how many times a number is multiplied by itself. A handful of rules let you simplify powers quickly, with numbers or with letters, as long as the base is the same.
Part 1 of 3: Learn it
In short
- Multiply: add the indices. Divide: subtract the indices.
- For a power raised to another power, the indices multiply.
- a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) is the nth root of a.
Where this is in your specification
Spec points: DfE N6, N7 and A4 (fractional indices are Higher tier)
| Board | Topic: Powers, roots and standard form |
|---|---|
| DfE content | N6, N7, N9 (and A4 for indices with letters) |
| AQA 8300 | Number |
| Edexcel 1MA1 | Number |
| OCR J560 | 3 Indices and surds; 1 Number operations and integers |
| Eduqas C300 | HN6, HN7, HN9 |
| Cambridge IGCSE 0580 | E1.3, E1.7, E1.8, E2.4 |
| National 5 C847 75 | Numerical skills: indices, scientific notation |
The rules
| Rule | In symbols | Example |
|---|---|---|
| Multiplying | aᵐ × aⁿ = aᵐ⁺ⁿ | x⁴ × x³ = x⁷ |
| Dividing | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁹ ÷ 5² = 5⁷ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (y²)⁵ = y¹⁰ |
| Power zero | a⁰ = 1 | 17⁰ = 1 |
| Negative power | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
The rules only work when the base is the same. 2³ × 3² cannot be combined into one power.
Numbers and letters together
Deal with the numbers and the letters separately. In 3x²y × 4x⁵y³, multiply 3 × 4 = 12, then x² × x⁵ = x⁷ and y × y³ = y⁴, giving 12x⁷y⁴. When a bracket is raised to a power, every part inside is raised to it: (2a³)⁴ = 16a¹².
Fractional indices (Higher)
A power of 1/n means the nth root: 49^(1/2) = √49 = 7 and 27^(1/3) = ∛27 = 3.
For a power like m/n, take the root first and then the power, because the numbers stay smaller: 8^(2/3) = (∛8)² = 2² = 4.
Simplify y⁸ ÷ y³ × y.
Show the answer
y⁸⁻³⁺¹ = y⁶.
Part 2 of 3: See it worked
Worked examples
Example 1
Simplify (2p³)² × 5p⁴.
- (2p³)² = 4p⁶
- 4 × 5 = 20
- p⁶ × p⁴ = p¹⁰
Answer: 20p¹⁰.
Example 2
Work out the value of 16^(−3/4).
- The negative power flips it: 1 ÷ 16^(3/4)
- Fourth root of 16 is 2
- 2³ = 8
Answer: 1/8.
Common mistakes
- Multiplying the indices when you should add them: x³ × x⁴ is x⁷, not x¹².
- Thinking a⁰ = 0. Any non-zero number to the power 0 is 1.
- Treating a negative power as a negative number: 3⁻² is 1/9, not −9.
- Forgetting to apply the power to the number in a bracket: (3x)² is 9x², not 3x².
What fraction is 4⁻³ equal to?
Show the answer
1/4³ = 1/64.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Simplify y⁸ ÷ y³ × y.
y⁸⁻³⁺¹ = y⁶.
What fraction is 4⁻³ equal to?
1/4³ = 1/64.
Work out 125^(2/3).
Cube root of 125 is 5, and 5² = 25.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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