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Improve Tuition · Foundation Maths · Year 6–10
Number · Indices & Powers

Indices & Powers

From index notation and the laws of indices to zero, negative powers and powers in real life.

6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure

An index, or power, is shorthand for repeated multiplication — 24 means 2 × 2 × 2 × 2 — and three rules cover most of the topic: add the powers when multiplying, subtract when dividing, and multiply for a power of a power. Pupils most often mix those rules up, multiply the base numbers when they should add the powers, or get stuck on what a zero or negative power means.

At Improve Tuition, a qualified teacher anchors each rule in why it works before drilling it, in small worked steps — with read-aloud, comfort spacing and a reading tint for pupils who take in maths more easily that way.

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Lesson 1

Index notation

Aim: Read and work out a power: the base and the index.

A power like a3 means “a multiplied by itself 3 times”: a × a × a. The big number is the base; the small raised number is the index (or power). The index counts how many copies are multiplied — it does not mean “times the base”.

2×2×2×2= 16
24 is four 2s multiplied — the small 4 counts the copies, so it is 16 (not 2 × 4).
Worked example
Work out 24.
  1. The big number is what you multiply; the small number says how many copies of the base are multiplied. So 24 means four 2s multiplied together.
  2. Write them all out so nothing is hidden: 2 × 2 × 2 × 2.
  3. Multiply two at a time, carrying the answer along. 2 × 2 = 4.
  4. 4 × 2 = 8.
  5. 8 × 2 = 16. So 24 = 16.
Watch out
24 is not 2 × 4 = 8. It is four 2s multiplied: 16. Also a1 = a.

Try one

Work out 33.

Show answer
33 means 3 × 3 × 3.
3 × 3 = 9.
9 × 3 = 27.
Check: that is three 3s, not 3 × 3 = 9. ✓
Practise this now — Section A →
Lesson 2

Multiplying powers

Aim: Multiply powers of the same base by adding the indices.

When the base is the same, multiplying powers just stacks more copies, so you add the indices: am × an = am+n.

aaa×aaaa→a73 a’s and 4 more a’s = 7
3 a’s and 4 more a’s make 7 a’s — so multiplying means you add the powers.
Worked example
Simplify a3 × a2.
  1. Why do we add? Check it with easy numbers first: 23 × 22 = 8 × 4 = 32, and 25 = 32 as well — the same answer.
  2. So multiplying powers of the same number is the same as adding the powers (here 3 + 2 = 5).
  3. Our question is the same idea, just with the letter a instead of 2.
  4. Add the powers the same way: 3 + 2 = 5.
  5. So a3 × a2 = a5.
Watch out
Add the indices, do not multiply them. And it only works when the base is the same.

Try one

Simplify a5 × a2. Find the power.

Show answer
The base a is the same, so add the indices.
5 + 2 = 7.
So a5 × a2 = a7 — the power is 7.
Check: 5 copies and 2 more copies makes 7 copies altogether. ✓
Practise this now — Section B →
Lesson 3

Dividing powers

Aim: Divide powers of the same base by subtracting the indices.

Dividing cancels copies top and bottom, so you subtract the indices: am ÷ an = am−n.

aaaaaaa→a4cancel 34 left
Start with 7 a’s, cancel 3 (crossed out), and 4 are left — so dividing means you subtract.
Worked example
Simplify a5 ÷ a2.
  1. Why subtract? Check with numbers: 25 ÷ 22 = 32 ÷ 4 = 8, and 23 = 8 — the same answer.
  2. So dividing powers of the same number is the same as subtracting the powers (here 5 − 2 = 3).
  3. Our question is the same idea, just with the letter a instead of 2.
  4. Subtract the powers the same way: 5 − 2 = 3.
  5. So a5 ÷ a2 = a3.
Watch out
Subtract the bottom index from the top one. Same base only.

Try one

Simplify a9 ÷ a4. Find the power.

Show answer
The base a is the same, so subtract the indices.
9 − 4 = 5.
So a9 ÷ a4 = a5 — the power is 5.
Check: 9 copies, cancel 4, leaves 5. ✓
Practise this now — Section C →
Lesson 4

Power of a power

Aim: Raise a power to another power by multiplying the indices.

A power raised to a power means repeating the whole thing, so you multiply the indices: (am)n = am×n.

aaaaaa3 groups of 2 → 2 × 3 = 6
(a2)3 is 3 groups of two a’s — that is 2 × 3 = 6, so you multiply the powers.
Worked example
Simplify (a2)3.
  1. Why multiply? Check with numbers: (22)3 = 43 = 64, and 26 = 64 — the same answer.
  2. So a power raised to a power is the same as multiplying the powers (here 2 × 3 = 6).
  3. Now our question. The small 3 means use the bracket three times.
  4. Multiply the powers: 2 × 3 = 6.
  5. So (a2)3 = a6.
Watch out
Here you multiply the indices. Compare with Lesson 2, where you add.

Try one

Simplify (a4)2. Find the power.

Show answer
A power raised to a power — multiply the indices.
4 × 2 = 8.
So (a4)2 = a8 — the power is 8.
Check: a4 × a4 adds to 8. ✓
Practise this now — Section D →
Lesson 5

Zero and negative indices

Aim: Use the rules a0 = 1 and a−n = 1/an.

Dividing a power by itself gives 1, and the rule says a0, so a0 = 1. Keep subtracting past zero and the index goes negative: a−n means 1 ÷ an.

a2=a × a÷aa1=a÷aa0=1÷aa−1=1/a÷aa−2=1/(a × a)
Each step down divides by a. At a0 you reach 1; below that the powers turn negative — a fraction.
Worked example
Simplify a2 ÷ a5. Find the power.
  1. Dividing again, so we subtract the powers.
  2. Top power take away bottom power: 2 − 5 = −3.
  3. What does a negative power mean? Check with numbers: 22 ÷ 25 = 4 ÷ 32, which is a fraction (less than 1).
  4. And 2−3 means 1 ÷ 23 — the same small fraction. So a negative power is just a fraction, not a negative answer.
  5. The power here is −3: a2 ÷ a5 = a−3, which also means 1/a3.
Watch out
a0 = 1, never 0. A negative index does not make the answer negative — it means a fraction.

Try one

Simplify a3 ÷ a7. Find the power.

Show answer
Subtract the indices: 3 − 7.
3 − 7 = −4.
So a3 ÷ a7 = a−4 — the power is −4.
Check: there are more on the bottom, so the index is negative. ✓
Practise this now — Section E →
Lesson 6

Mixing the laws

Aim: Use more than one index law in the same question, and in context.

Work left to right, using each law in turn: add when multiplying, subtract when dividing, multiply for a power of a power. In shapes, area multiplies two lengths and volume multiplies three.

×→add the powers÷→subtract the powers(power)n→multiply the powers
The three rules together: × → add, ÷ → subtract, a power of a power → multiply.
Worked example
Simplify a5 × a2 ÷ a3. Find the power.
  1. This has two rules in it, so do one step at a time, left to right.
  2. Step 1 — the multiply: a5 × a2. Multiplying means add: 5 + 2 = 7. That gives a7.
  3. Step 2 — the divide: a7 ÷ a3. Dividing means subtract: 7 − 3 = 4.
  4. So the final answer is a4 — the power is 4.
Watch out
Do one law at a time. Add for ×, subtract for ÷, multiply for a bracket power.

Try one

A cube has edges of length a2. Its volume is (a2)3. Find the power.

Show answer
Volume of a cube = side × side × side, which is (a2)3.
Power of a power — multiply the indices: 2 × 3 = 6.
So the volume is a6 — the power is 6.
Check: a2 × a2 × a2 adds to 6. ✓
Practise this now — Section F →
↑ Back to the lessons
Section A · Index notation
LEARNING OBJECTIVE · GRADE 1 · CALCULATION

Read and evaluate a power.

Success criteria — I can:
  • the base is multiplied by itself
  • the index counts the copies
  • multiply step by step
1.Work out 24.
Answer:
2.Work out 52.
Answer:
3.Work out 33.
Answer:
4.Work out 103.
Answer:
5.Work out 26.
Answer:
↑ Back to the lessons
Section B · Multiplying powers
LEARNING OBJECTIVE · GRADE 3 · CALCULATION

Multiply powers of the same base.

Success criteria — I can:
  • check the base is the same
  • add the indices
  • state the new power
6.a3 × a4 = a? — find the power.
Answer:
7.a5 × a2 = a? — find the power.
Answer:
8.a6 × a3 = a? — find the power.
Answer:
9.a2 × a8 = a? — find the power.
Answer:
10.a4 × a4 = a? — find the power.
Answer:
↑ Back to the lessons
Section C · Dividing powers
LEARNING OBJECTIVE · GRADE 3 · CALCULATION

Divide powers of the same base.

Success criteria — I can:
  • check the base is the same
  • subtract the indices
  • state the new power
11.a7 ÷ a3 = a? — find the power.
Answer:
12.a9 ÷ a4 = a? — find the power.
Answer:
13.a8 ÷ a2 = a? — find the power.
Answer:
14.a10 ÷ a6 = a? — find the power.
Answer:
15.a6 ÷ a1 = a? — find the power.
Answer:
↑ Back to the lessons
Section D · Power of a power, zero and negative indices
LEARNING OBJECTIVE · GRADE 4 · CALCULATION

Use power-of-a-power, zero and negative indices.

Success criteria — I can:
  • multiply for a power of a power
  • remember a⁰ = 1
  • let the index go negative
16.(a2)3 = a? — find the power.
Answer:
17.(a4)2 = a? — find the power.
Answer:
18.a4 ÷ a4 = a? — find the power.
Answer:
19.Work out 70.
Answer:
20.a2 ÷ a5 = a? — find the power.
Answer:
↑ Back to the lessons
Section E · Powers in shapes
LEARNING OBJECTIVE · GRADE 3 · WORD PROBLEM

Use indices for areas and volumes.

Success criteria — I can:
  • area multiplies two lengths
  • volume multiplies three
  • add the indices
21.A square has sides of length a3. Its area is a3 × a3 = a?. Find the power.
Answer:
22.A rectangle is a5 long and a2 wide. Its area is a?. Find the power.
Answer:
23.A number is written 25. Work out its value.
Answer:
24.A square floor tile has side 5 cm. Its area is 52 cm². Work out the area.
Answer:
25.A small cube has edges of length a1 . Three of these lengths multiply for its volume: a1 × a1 × a1 = a?. Find the power.
Answer:
↑ Back to the lessons
Section F · Powers in real life
LEARNING OBJECTIVE · GRADE 3 · WORD PROBLEM

Evaluate powers in growth and counting problems.

Success criteria — I can:
  • spot the repeated multiplying
  • write it as a power
  • work out the value
26.A single cell divides into 2 every hour. Starting from 1 cell, after 6 hours there are 26 cells. How many cells is that?
Answer:
27.A piece of paper doubles in thickness each fold. After 10 folds it is 210 times as thick. Work out 210.
Answer:
28.A PIN uses 3 digits, each from 0–9, giving 103 possible codes. How many codes is that?
Answer:
29.Bacteria triple every hour. From 1 bacterium, after 4 hours there are 34. Work out 34.
Answer:
30.A cube has edges of 2 cm. Its volume is 23 cm³. Work out the volume.
Answer:
↑ Back to the lessons
Section G · Mixing the laws
LEARNING OBJECTIVE · GRADE 4 · WORD PROBLEM

Combine more than one index law.

Success criteria — I can:
  • one law at a time
  • add, subtract or multiply the indices
  • state the final power
31.Simplify a5 × a2 ÷ a3. Find the power.
Answer:
32.Simplify (a2)3 × a4. Find the power.
Answer:
33.Simplify a8 ÷ a2 ÷ a3. Find the power.
Answer:
34.A cube has side 22 cm. Its volume is (22)3 = 2?. Find the power.
Answer:
35.Simplify a3 × a3 × a2. Find the power.
Answer:
0 of 35 answered
Guide

What are the laws of indices?

An index (plural indices), or power, tells you how many times to multiply a number by itself: 53 means 5 × 5 × 5. The large number is the base and the small raised number is the index.

Three rules do most of the work: when multiplying powers of the same base, add the indices (am × an = am+n); when dividing, subtract them; and for a power of a power, multiply them. Also, anything to the power 0 equals 1, and a negative index means ‘one over’ the positive power.

Common mistakes. Multiplying the bases instead of adding the indices, applying the rules to powers with different bases, or misreading a0 as 0 rather than 1.

Common questions
What is an index or power?
It is shorthand for repeated multiplication. In 43, the base 4 is multiplied by itself three times: 4 × 4 × 4 = 64.
What is the rule for multiplying indices?
When the base is the same, add the indices: am × an = am+n. For example, 23 × 24 = 27.
What is the rule for dividing indices?
When the base is the same, subtract the indices: am ÷ an = am−n. For example, 56 ÷ 52 = 54.
What does a power of 0 mean?
Any non-zero number to the power 0 equals 1. For example, 70 = 1.
What does a negative power mean?
A negative index means one divided by the positive power. For example, 2−3 = 1 ÷ 23 = 1/8.
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