Laws of indices: multiplying, dividing, negative and fractional

What are indices in maths?

An index (plural indices) is the small raised number that says how many times a number is multiplied by itself. In 5³, the 5 is the base and the 3 is the index. 5³ means 5 × 5 × 5, which is 125. Indices are also called powers or exponents, so 5³ is "5 to the power 3" or "5 cubed".

Children meet the first two at primary school. In Year 5 maths they learn square numbers and cube numbers and the notation for them: 4² is 16 and 2³ is 8. The square numbers run down the diagonal of the times table grid. At key stage 3, the same notation is used with letters: a² means a × a.

What are the laws of indices?

The laws of indices are three main rules, plus what a zero index means. They only work when the bases are the same.

Law	Rule	Example
Multiplying	add the indices	32 × 35 is 37
Dividing	subtract the indices	56 ÷ 52 is 54
Power of a power	multiply the indices	(23)2 is 26
Zero index	anything (except 0) to the power 0 is 1	70 is 1

With letters, the same rules read: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n and (a^m)ⁿ = a^mn.

How to add indices: multiplying powers

You add the indices when you multiply two powers of the same number. Write it out and you can see why:

2³ × 2⁴ is (2 × 2 × 2) × (2 × 2 × 2 × 2). That's seven 2s multiplied together, so it's 2⁷, and 3 + 4 = 7.

The same works with letters: x⁴ × x⁵ is x⁹. With numbers in front, multiply the numbers and add the indices: 3x² × 4x³ is 12x⁵.

Adding two powers is different. 2³ + 2⁴ is not 2⁷. Work each one out first: 8 + 16 = 24.

Dividing powers and powers of powers

You subtract the indices when you divide. 5⁶ ÷ 5² is six 5s multiplied together, divided by two 5s. Two pairs cancel, which leaves four 5s: 5⁴, which is 625. And 6 − 2 = 4.

Worked example: a power of a power

(3²)³ means 3² × 3² × 3². Using the multiplying law, that's 3 to the power 2 + 2 + 2, which is 3⁶. Multiplying the indices gets there faster: 2 × 3 = 6. And 3⁶ is 729.

Why anything to the power 0 is 1

Divide a power by itself: 4³ ÷ 4³ is 1, because any number divided by itself is 1. But the dividing law says it's 4⁰, because 3 − 3 = 0. So 4⁰ has to be 1.

How to do negative indices

A negative index means one divided by the power. 2⁻³ means 1 ÷ 2³, which is ¹⁄₈. It is not a negative number.

You can see this by carrying on a pattern. 2³ is 8, 2² is 4, 2¹ is 2, 2⁰ is 1. Each step halves, so the next ones are 2⁻¹, which is ¹⁄₂, then 2⁻², which is ¹⁄₄.

Negative powers of 10 give the decimal places: 10⁻¹ is 0.1, 10⁻² is 0.01 and 10⁻³ is 0.001. That's what makes standard form work for very small numbers. The decimals guide covers tenths, hundredths and thousandths.

The laws still work with negative indices: 5² × 5⁻³ is 5⁻¹, which is ¹⁄₅.

How to do fractional indices

A fractional index means a root. An index of ¹⁄₂ is a square root, and an index of ¹⁄₃ is a cube root.

• 16 to the power ¹⁄₂ is the square root of 16, which is 4, because 4 × 4 = 16.
• 27 to the power ¹⁄₃ is the cube root of 27, which is 3, because 3 × 3 × 3 is 27.

Worked example: 8 to the power ²⁄₃

With a fraction like ²⁄₃, the bottom number is the root and the top number is the power. Take the cube root of 8, which is 2. Then square it: 2² is 4. So 8 to the power ²⁄₃ is 4.

Fractional indices make sense with the multiplying law. 9 to the power ¹⁄₂, times itself, is 9 to the power ¹⁄₂ + ¹⁄₂, which is 9¹. The number that multiplies by itself to make 9 is 3, the square root.

In the key stage 4 programme of study, fractional indices are extra content for more highly attaining pupils, so not every GCSE student meets them.

Common mistakes with the laws of indices

• Multiplying the indices instead of adding. 2³ × 2⁴ is 2⁷, not 2¹².
• Multiplying the bases. 2³ × 2⁴ is not 4⁷. The base stays the same.
• Using the laws with different bases. 2³ × 3² can't be simplified with the laws: work it out as 8 × 9 = 72.
• Reading 2³ as 2 × 3. 2³ is 2 × 2 × 2, which is 8, not 6.
• Thinking a zero index gives 0. 7⁰ is 1.
• Thinking a negative index makes a negative number. 2⁻³ is ¹⁄₈, a small positive number.

How to help at home

• Ask for the long version. When your child writes 3⁴, ask them to write it out as 3 × 3 × 3 × 3. Most of the laws are easy to see once it's written out.
• Know the first few powers. Squares up to 12², cubes up to 5³, and powers of 2 up to 2¹⁰ (1,024) come up again and again. The key stage 3 programme of study expects children to recognise powers of 2, 3, 4 and 5.
• Play "which law?" Write a few calculations and ask whether to add, subtract or multiply the indices.
• Ask the school about the order. Key stage 3 covers Years 7 to 9 and schools decide when each topic is taught. Their scheme of work or curriculum map shows when indices come up.

Where the laws of indices fit

Square and cube numbers start in Year 5. Key stage 3 brings in powers and roots more generally, along with powers in the order of operations and in algebra basics. The laws of indices are named in the key stage 4 programme of study, for GCSE in Years 10 and 11, where children use them to simplify expressions. Index notation also turns up in prime factors: 72 is 2³ × 3². For the other topics in these years, see the KS3 maths guide.

Indices questions to try

1. Work out 3⁴.
2. Simplify 7⁵ × 7³, leaving your answer as a power of 7.
3. Simplify x⁹ ÷ x⁴.
4. Simplify (5²)⁴, leaving your answer as a power of 5.
5. Work out 10⁻² as a fraction and as a decimal.
6. Work out 25 to the power ¹⁄₂.

Answers

1. 81 (3 × 3 × 3 × 3)
2. 7⁸ (add the indices)
3. x⁵ (subtract the indices)
4. 5⁸ (multiply the indices)
5. ¹⁄₁₀₀, which is 0.01
6. 5, the square root of 25