An index (or power) is a shorthand for repeated multiplication of the same number by itself. Instead of writing 3 × 3 × 3 we write 3³, and read it as 'three to the power of three' or 'three cubed'. The big number at the bottom is called the base, and the small raised number is the index (or exponent). In Year 7 pupils meet this topic formally for the first time, and also learn the inverse operation — the square root. This guide explains to parents exactly what your child is learning, which laws they need to remember, and how to help them practise at home without getting tangled up.
An index (or power) is a shorthand for repeated multiplication of the same number by itself. Instead of writing 3 × 3 × 3 we write 3³, and read it as 'three to the power of three' or 'three cubed'. The big number at the bottom is called the base, and the small raised number is the index. In Year 7 pupils meet this topic formally for the first time, and also learn the inverse operation — the square root. This guide explains to parents exactly what your child is learning, which laws they need to remember, and how to help them practise at home without getting tangled up.
What is an index?
An index (plural: indices; also called a power or exponent) is a short way of writing a number multiplied by itself a number of times. The expression 3² means 3 × 3 = 9, and the expression 3³ means 3 × 3 × 3 = 27. The number 3 is the base — the number being multiplied by itself. The small number written above and to the right of it is the index — it tells you how many times to multiply the base by itself.
The two most common cases are the second power and the third power. The second power (a²) is called 'squared', because it is linked to the area of a square. The third power (a³) is called 'cubed', because it is linked to the volume of a cube. For example: 4² = 16, and 4³ = 64.
A Year 7 pupil should know the following square numbers by heart: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100, 11² = 121, 12² = 144. (Square numbers up to 12² are actually introduced in Year 5 and 6, so this should be revision.) Memorising this table saves precious time in tests and prevents calculation errors. It is worth pinning it on the fridge for the first couple of months of the topic.
Two important notes on what the symbol means: 0 to any positive power equals 0 (because 0 × 0 = 0), and any number to the power of 1 equals itself (5¹ = 5). In Year 7 pupils usually start with positive whole-number indices (1, 2, 3 …), and the definition is gradually widened later.
The basic index laws
In Year 7 pupils learn three or four central laws that come up again and again in questions. A good understanding of them saves unnecessary long calculations.
The multiplication law: when multiplying two powers with the same base, add the indices. The formula: aᵐ × aⁿ = aᵐ⁺ⁿ. Example: 2³ × 2⁴ = 2⁷ = 128. You can see it directly: (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 × 2.
The division law: when dividing two powers with the same base, subtract the indices. The formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Example: 5⁵ ÷ 5² = 5³ = 125.
The power-of-a-power law: when raising a power to a power, multiply the indices. The formula: (aᵐ)ⁿ = aᵐ·ⁿ. Example: (3²)³ = 3⁶ = 729. Notice — you do not add 2 + 3, you multiply 2 × 3.
A power of a product: (a·b)ⁿ = aⁿ · bⁿ. For example: (2 × 5)² = 10² = 100, and also 2² × 5² = 4 × 25 = 100. The same result.
The power of 0 and the power of 1: these two cases confuse many pupils. The rule: any number (except 0) to the power of 0 equals 1. So 5⁰ = 1, 100⁰ = 1, and also 1⁰ = 1. Why? Because 5³ ÷ 5³ = 5⁰ on one hand, and equals 1 on the other (any number divided by itself is 1). This explanation gives the law an inner logic, not just something to memorise.
Any number to the power of 1 equals itself: 7¹ = 7, 250¹ = 250. It seems obvious, but it is important to say explicitly, because in simplifying questions pupils sometimes forget that a single variable like x is really x¹.
Roots — the inverse of a power
A square root is the inverse operation of squaring. If 5² = 25, then the square root of 25 is 5. The notation: √25 = 5. The question a square root asks is: 'Which positive number, when squared, gives the number under the sign?'
In Year 7 the work is mainly with exact roots — numbers whose square root is a nice whole number. A pupil should know the following list by heart: √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √144 = 12. Notice — this is exactly the reverse of the table of square numbers we mentioned earlier.
Numbers that are not perfect squares (for example 2, 3, 5, 7, 10) do have a square root, but it is not a whole number. √2 ≈ 1.414, and √10 ≈ 3.162. In Year 7 the answer is usually left with the root sign (write √2 rather than the decimal value), and in other cases a calculator is used.
One point worth clarifying: at school the square root is always defined as the positive number. It is true that (−5)² = 25 as well, but the notation √25 refers only to the positive solution, that is 5. The second part is studied in detail in Years 9–10 when quadratic equations begin.
A last idea for later: negative indices. Year 7 only touches on this lightly, and Year 8 studies it in depth. The rule is a⁻ⁿ = 1/aⁿ. For example: 2⁻¹ = 1/2, and 2⁻³ = 1/8. If your child has not yet seen this — there is no need to go into it.
Powers in real life — where do you see them?
Powers are not just a dry exercise in a textbook. They appear everywhere area, volume or rapid growth is measured.
The area of a square: the formula is s², where s is the side length. A square with a side of 7 cm has an area of 49 cm². This is perhaps the most intuitive use of powers a Year 7 pupil meets.
The volume of a cube: the formula is s³. A cube with a side of 4 cm has a volume of 64 cm³. Here you can see clearly why the third power is called 'cubed'.
Pythagoras' theorem: in a right-angled triangle, a² + b² = c². This is one of the central topics of Years 8–9, and it uses powers and roots together — you square, then take a square root to find a missing side.
Computers and memory: a kilobyte is about 2¹⁰ = 1024 bytes, and a megabyte is 2²⁰ bytes. The whole structure of digital memory is built on powers of 2.
Compound interest in a savings account: if you deposit £1,000 at 5% a year, after 10 years the account will hold 1000 × (1.05)¹⁰ pounds. The compound interest formula is one of the main financial reasons to learn powers properly.
5 tips for practising at home
- **Learn the square numbers up to 12 by heart.** Print the table and pin it somewhere prominent — on the fridge, above the desk, or next to the mirror. Memorising just 12 values saves a lot of time in tests and prevents repeated calculation errors.
- **Practise 10 minutes a day, not an hour a week.** Maths is a skill that strengthens gradually. Short daily practice beats a weekly marathon. Ten index questions a day is a good rate for it to sink in.
- **Use focused worksheets.** An indices worksheet for Year 7 (printable PDF) includes questions graded from easy to hard, with an answer key. You can print it and work with a pencil — sometimes that is more effective than a screen.
- **Mix in game-mode practice.** Children who find it hard to concentrate on written work often respond brilliantly to interactive practice. Practising indices in game mode turns revision into something to look forward to, not run away from.
- **Do a summary mock before a test.** A week before the test, run a timed practice test under real conditions — a set time limit, no distractions, no phone. It reveals exactly where the gaps in knowledge are and what needs strengthening.
Common mistakes
Three mistakes come up again and again with Year 7 pupils, and it is worth talking about them in advance so your child can spot them for themselves.
Mistake 1: 3² = 6. This is an especially common mistake at the start of the topic. Pupils confuse a power with multiplication and work out 3 × 2 = 6 instead of 3 × 3 = 9. The fix: always work in two steps — first write the power out as a product (3 × 3), and only then calculate.
Mistake 2: √16 = 8. The mirror-image mistake of the one above: dividing 16 by 2 instead of asking 'which number squared gives 16?'. The correct answer is 4 (because 4² = 16). The best help is going over the table of square numbers again and again.
Mistake 3: mixing up 2³ and 3². The two expressions look similar, but the results are completely different: 2³ = 2 × 2 × 2 = 8, whereas 3² = 3 × 3 = 9. Always identify which is the base (the big number at the bottom) and which is the index (the small number at the top).
Mistake 4: adding indices in a power of a power. Pupils write (2³)² = 2⁵ instead of 2⁶. The rule: in a power of a power you multiply the indices, not add them.
Frequently asked questions
At what age are indices first taught?
Early ideas of powers (mainly square and cube numbers, in the context of area and volume) appear in Years 5–6 of the national curriculum. The formal study of the index laws, full mathematical notation, powers other than two and square roots begins in Year 7 at KS3. Year 8 extends to negative indices, Year 9 to fractional indices, and GCSE to standard form, surds and exponential growth.
What is the difference between 2³ and 3²?
The difference is fundamental. 2³ means 2 × 2 × 2 = 8, that is the number 2 multiplied by itself 3 times. Whereas 3² means 3 × 3 = 9, that is the number 3 multiplied by itself 2 times. The base and the index swap roles — and that completely changes the result.
Why is any number to the power of 0 equal to 1?
The logical explanation: look at the division law. 5³ ÷ 5³ should be 5⁰ by the law (subtract the indices: 3 − 3 = 0). On the other hand, any number divided by itself is 1. So 5⁰ must be 1. The same explanation works for any base other than 0 (the one exception: 0⁰ is undefined).
Is a calculator allowed in the test?
It depends on the teacher and the type of test. In many class tests a calculator is not allowed, so it is essential that your child knows the table of square numbers and the common exact roots by heart. In end-of-year KS3 assessments and later at GCSE, one of the papers is always non-calculator, so quick mental calculation saves precious time.
How do you practise indices if you cannot remember them?
The advice is to go back to basics. If your child cannot remember 7², have them write 7 × 7 and work it out. The more they repeat this, the stronger the memory becomes. You can also use flashcards: on one side 8², on the other 64. Five minutes every evening gives complete recall within a fortnight.
Do indices appear at GCSE?
Yes, and centrally. Indices are a basic topic that accompanies a pupil from Year 7 to GCSE. At GCSE you meet the index laws with negative and fractional indices, standard form, surds and compound interest calculations — and all of them are built on the foundations learnt in Year 7. A good investment now saves a lot of effort later.
What is a negative index?
A negative index is a shorthand for 1 divided by the positive power. The rule: a⁻ⁿ = 1/aⁿ. For example: 2⁻¹ = 1/2, and 2⁻³ = 1/8. This topic is studied in depth in Year 8, but is sometimes touched on towards the end of Year 7. There is no need to hurry — a good understanding of positive indices is a necessary condition first.
How do I help a child who gets confused?
First, do not do the question for them. Let them find the mistake themselves: 'Let's write 3² as a product — what do we get?'. Second, go over the table of square numbers once a day, even when it seems they already know it. Third, use links to real life — the area of a room, the volume of a box. Fourth, if the difficulty lasts beyond two or three weeks, read about further ways to help in our other articles or arrange a few focused sessions with a tutor.
Graded questions with full solutions — free to print
Download a worksheet — indices for Year 7 ←