The laws of indices — every rule with examples
What are the laws of indices?
A power is a short way of writing repeated multiplication of the same number: 2⁵ means 2 multiplied by itself 5 times. The laws of indices are shortcuts that save you writing out that long multiplication — instead of expanding every power into a product, you simply add or subtract the indices. In this guide we go through every rule one at a time, with a full example for each.
Worked examples, step by step
Work out 2³ · 2⁴
- When the base is the same (2 in both), you multiply powers by adding the indices
- 2³ · 2⁴ = 2³⁺⁴ = 2⁷
- Check by expanding: 2³ = 8 and 2⁴ = 16, and 8 × 16 = 128
- Answer: 2⁷ = 128
Work out 3⁵ ÷ 3²
- When the base is the same, you divide powers by subtracting the indices
- 3⁵ ÷ 3² = 3⁵⁻² = 3³
- Check by expanding: 3⁵ = 243 and 3² = 9, and 243 ÷ 9 = 27
- Answer: 3³ = 27
Work out (3²)⁴
- With a power raised to a power, multiply the indices together: 2 × 4 = 8
- So (3²)⁴ = 3⁸
- Partial check: 3² = 9, and 9⁴ = 6561, and 3⁸ = 6561 — exactly the same result
- Answer: 3⁸ = 6561
Work out 5⁻²
- A negative index means "turn it into a fraction": move the base to the denominator and the index becomes positive
- 5⁻² = 1 ÷ 5²
- Work out the denominator: 5² = 25
- Answer: 5⁻² = 1 ÷ 25 = 0.04
Now try it yourself
Frequently asked questions
Why is any number to the power 0 equal to 1?
You can see it from the division law: xᵃ ÷ xᵃ is always 1 (any number divided by itself), and by the subtraction rule for indices xᵃ ÷ xᵃ = xᵃ⁻ᵃ = x⁰. Both routes give the same expression, so x⁰ = 1 for every base other than zero.
What is a negative index?
A negative index turns the base into a fraction: x⁻ⁿ equals 1 divided by xⁿ. For example 2⁻³ equals 1 divided by 8, that is one eighth. The larger the negative index is in size, the smaller the number.
What do you do when the bases are different?
The multiplication and division laws (adding and subtracting indices) only work when the base is the same on both sides. If the bases differ but the index is the same — for example 2³ · 5³ — you can use the power-of-a-product law: aⁿ · bⁿ = (a · b)ⁿ. If both the base and the index differ, you usually have to work out each power separately.
How are roots connected to indices?
A root is a power with a fractional index: the square root of x equals x to the power one half, and the cube root equals x to the power one third. Because of that, every law of indices — including adding and subtracting indices — works on roots too. Fractional indices are Higher tier at GCSE.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specificationsLast updated: