Chapter 2 · Foundation tier

GCSE Maths Factors, Multiples, Primes and Powers (Foundation): worked examples and questions with answers

By the end of this chapter you will be able to

List multiples and factors and use divisibility tests
Recognise prime numbers and write a number as a product of prime factors
Find the HCF and LCM using lists, prime factors and Venn diagrams
List outcomes systematically and use the product rule for counting
Use squares, cubes and roots, and recognise the powers of 2, 3, 4 and 5
Use the laws of indices, including zero and negative indices
Estimate roots and powers, and use a calculator for them

You should already know

Times tables up to 12 × 12
Multiplying and dividing whole numbers (Chapter 1)
The order of operations, BIDMAS (Chapter 1)

Where this leads

Simplifying fractions and adding fractions (Chapter 3)
Standard form, fractional indices and surds (Chapter 6)
Index laws in algebra (Chapter 7)
Counting outcomes in probability (Chapter 23)

Edexcel GCSE (9–1) Mathematics 1MA1, Foundation tier. Content references: N3, N4, N5, N6, N7.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

2.1 Multiples and factors: Worksheet (PDF) · Answers (PDF) · lesson page
2.2 Prime numbers: Worksheet (PDF) · Answers (PDF) · lesson page
2.3 Prime factorisation: Worksheet (PDF) · Answers (PDF) · lesson page
2.4 Highest common factor and lowest common multiple: Worksheet (PDF) · Answers (PDF) · lesson page
2.5 Systematic listing and the product rule: Worksheet (PDF) · Answers (PDF) · lesson page
2.6 Squares, cubes, roots and powers: Worksheet (PDF) · Answers (PDF) · lesson page
2.7 Powers and the laws of indices: Worksheet (PDF) · Answers (PDF) · lesson page
2.8 Zero and negative indices: Worksheet (PDF) · Answers (PDF) · lesson page
2.9 Estimating roots and powers, and using a calculator: Worksheet (PDF) · Answers (PDF) · lesson page
Check-up and exam-style questions: Worksheet (PDF) · Answers (PDF) · lesson page

Exercise P Do you remember?

Non-calculator

FWork out 8 × 7.

Done

FWork out 132 ÷ 12.

Done

FWork out 3 + 4 × 5.

Done

FWrite 1218 in its simplest form.

Done

FWork out 4 × 4 × 4.

Done

2.1

Multiples and factors

Edexcel 1MA1: N3, N4
Grade 1-3

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Multiples and factorsComing soon

Look at the 3 times table: 3, 6, 9, 12, 15 ... These numbers are the multiples of 3. A multiple of a number is what you get when you multiply it by a whole number (1, 2, 3, 4 ...). Multiples go on for ever, and each one is bigger than the one before. The first multiple of any number is the number itself.

On a 100 square, the multiples of 3 are blue, the multiples of 4 are orange and the numbers that are in both lists (the common multiples 12, 24, 36 ...) are purple.

Factors work the other way round. A factor of a number is a whole number that divides into it exactly, with no remainder. Since 3 × 4 = 12, both 3 and 4 are factors of 12. We call 3 and 4 a factor pair of 12. To find all the factors of a number, look for all its factor pairs, starting with 1 × the number itself.

A rectangle of 12 dots can be made in three ways. Each way is a factor pair of 12: 1 × 12, 2 × 6 and 3 × 4. So the factors of 12 are 1, 2, 3, 4, 6 and 12.

A number can be a common factor of two numbers (it divides into both) or a common multiple (it is in both times tables). For example, 2, 3 and 6 are all common factors of 12 and 18. And 12, 24 and 36 are common multiples of 3 and 4.

Long division is slow, so we use divisibility tests to see quickly whether a number divides exactly.

Divisibility tests

2: the last digit is 0, 2, 4, 6 or 8.
3: the digits add up to a multiple of 3.
4: the last two digits make a multiple of 4.
5: the last digit is 0 or 5.
6: it passes the tests for both 2 and 3.
8: the last three digits make a multiple of 8.
9: the digits add up to a multiple of 9.
10: the last digit is 0.

Key ideas

A multiple of n is in the n times table: n, 2n, 3n ...
A factor of n divides into n exactly. Factors come in pairs.
Every number has 1 and itself as factors.
A common factor divides two numbers; a common multiple is in both times tables.
Use divisibility tests instead of long division.

Worked example 2.1.1

Write down the first six multiples of 7.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 7 are the numbers in the 7 times table. Multiply 7 by 1, 2, 3 ... in turn.

Remember what a multiple is. A multiple of 7 is what you get when you multiply 7 by a whole number.
Multiply 7 by 1, 2, 3, ... for 6 lines. Each multiple is 7 more than the one before it, so you are just counting up in 7s.
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
7 × 5 = 35
7 × 6 = 42
Write the multiples as a list. The question asks for the numbers only.
7, 14, 21, 28, 35, 42

Check. Count up in 7s: 7 → 14 → 21 → 28 → 35 → 42. Each step adds 7. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Worked example 2.1.2

Find all the factors of 36.

Method. Find the factor pairs of 36 (two numbers that multiply to give 36) and then list every number in those pairs.

Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
1 × 36 = 36
Test 2, 3, 4, ... in order. Stop at 6. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 6 because 6 × 6 = 36: the pairs have met in the middle.
1 × 36 = 36 ✓ pair: 1 and 36
2 × 18 = 36 ✓ pair: 2 and 18
3 × 12 = 36 ✓ pair: 3 and 12
4 × 9 = 36 ✓ pair: 4 and 9
5: 36 ÷ 5 = 7 remainder 1, so 5 is not a factor
6 × 6 = 36 ✓ pair: 6 and 6
Write every factor in order, smallest to largest. Collect both numbers from every pair.
1, 2, 3, 4, 6, 9, 12, 18, 36

Check. 5 pairs, and 6 is paired with itself, giving 9 factors. Each factor divides 36 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Worked example 2.1.3

Use a divisibility test to decide whether 4716 is divisible by 6.

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 6 instead of doing a long division.

Recall the test for 6. A number is divisible by 6 if it is divisible by both 2 and 3 (because 6 = 2 × 3).
Apply the test to the number. Use only the digits the test asks about.
Divisible by 2? The last digit is 6: yes.
Divisible by 3? Digit sum 4 + 7 + 1 + 6 = 18: yes.
Decide. If the test is passed the number is divisible, otherwise it is not.
Yes, 4716 is divisible by 6.

Check. 4716 ÷ 6 = 786 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Worked example 2.1.4

Which numbers in the list 12, 20, 24, 36, 40 are multiples of 4 and factors of 72?

Method. Test every number against both conditions, then keep the numbers that pass both.

List the factors of 72. We need to know what divides into 72. Use factor pairs.
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Test each number in the list: is it a multiple of 4? A multiple of 4 divides by 4 exactly.
12 ÷ 4 = 3
20 ÷ 4 = 5
24 ÷ 4 = 6
36 ÷ 4 = 9
40 ÷ 4 = 10
Test each number: is it a factor of 72? Look it up in the list of factors of 72.
12: in the list ✓
20: not in the list ✗
24: in the list ✓
36: in the list ✓
40: not in the list ✗
Keep the numbers that pass both tests. The question says “and”, so a number has to pass both.
12, 24, 36

Check. 12, 24, 36: each is divisible by 4 (12 = 4 × 3, 24 = 4 × 6, 36 = 4 × 9) and divides 72 (72 ÷ 12 = 6, 72 ÷ 24 = 3, 72 ÷ 36 = 2). ✓

Watch out. When a question says “multiple of ... and factor of ...” make one list for each condition.

Watch out

Multiples are bigger than (or equal to) the number; factors are smaller than (or equal to) it. 5 is a factor of 20 and 40 is a multiple of 20. Also, do not forget 1 and the number itself when you list factors.

2.2

Prime numbers

Edexcel 1MA1: N4
Grade 2-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Prime numbersComing soon

Some numbers have lots of factors. 12 has six: 1, 2, 3, 4, 6 and 12. Other numbers have very few. The number 7 can only be made by 1 × 7, so its only factors are 1 and 7. A number like this is called a prime number.

A prime number is a whole number that has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19 ... A number with more than two factors, such as 12, is called a composite number.

Two numbers cause trouble. 1 is not prime, because it has only one factor (1 itself). 2 is prime, and it is the only even prime, because every other even number is also divisible by 2, giving it a third factor.

The Sieve of Eratosthenes: cross out 1, then every multiple of 2 (except 2), every multiple of 3 (except 3), every multiple of 5 and 7. What is left, shaded orange, is the 25 primes up to 100.

To find the primes this way, shade 2 then cross out all its multiples, move to the next number not crossed out (3), shade it and cross out its multiples, and so on. Any number left must be prime. You should learn the primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

How can you test a larger number such as 91 without writing out all its factors? Factors come in pairs, one small and one large. If a number has a factor bigger than its square root, its partner is smaller than the square root. So we only need to test primes up to the square root. For 91, 9 × 9 = 81 and 10 × 10 = 100, so the square root is between 9 and 10. We test 2, 3, 5 and 7. Since 91 = 7 × 13, it is not prime.

Key ideas

A prime has exactly two factors: 1 and itself.
1 is not prime. 2 is the only even prime.
A number with more than two factors is composite.
Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
To test a number, divide by the primes up to its square root. If none goes in exactly, it is prime.

Worked example 2.2.1

Is 29 a prime number? Show your working.

Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.

Method. Test whether any prime up to √29 divides into 29.

Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.
To show 29 is not prime we need one extra factor.
Decide which numbers to test. Factors come in pairs. If 29 had a factor bigger than √29, its partner would be smaller than √29 and we would meet it first. Here 5 × 5 = 25 < 29 and 6 × 6 = 36 > 29, so we only need to test the primes up to 5. We test primes only, because any other divisor is built from primes.
Primes up to 5: 2, 3, 5
Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.
29 ÷ 2 = 14 remainder 1, so 2 is not a factor.
29 ÷ 3 = 9 remainder 2, so 3 is not a factor.
29 ÷ 5 = 5 remainder 4, so 5 is not a factor.
Decide. None of the primes up to 5 divides into 29, so 29 has no factors except 1 and itself.
29 is prime.

Check. Its only factors are 1 and 29. ✓

Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).

Worked example 2.2.2

Is 91 a prime number? Show your working.

Method. Test whether any prime up to √91 divides into 91.

Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.
To show 91 is not prime we need one extra factor.
Decide which numbers to test. Factors come in pairs. If 91 had a factor bigger than √91, its partner would be smaller than √91 and we would meet it first. Here 9 × 9 = 81 < 91 and 10 × 10 = 100 > 91, so we only need to test the primes up to 9. We test primes only, because any other divisor is built from primes.
Primes up to 9: 2, 3, 5, 7
Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.
91 ÷ 2 = 45 remainder 1, so 2 is not a factor.
91 ÷ 3 = 30 remainder 1, so 3 is not a factor.
91 ÷ 5 = 18 remainder 1, so 5 is not a factor.
91 ÷ 7 = 13 exactly, so 7 is a factor.
Decide. 7 is a factor other than 1 and 91, so there are more than two factors.
91 = 7 × 13, so 91 is not prime.

Check. 7 × 13 = 91. ✓

Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).

Worked example 2.2.3

Write down all the prime numbers between 40 and 60.

Method. Check each number from 40 to 60: if a small prime divides it, it is not prime.

List the numbers to check. Every number in the range must be tested.
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60
Test each one. Look for a factor other than 1 and the number itself (try 2, 3, 5, 7 ...).
40 = 2 × 20, not prime
41: no prime up to √41 divides it, so prime ✓
42 = 2 × 21, not prime
43: no prime up to √43 divides it, so prime ✓
44 = 2 × 22, not prime
45 = 3 × 15, not prime
46 = 2 × 23, not prime
47: no prime up to √47 divides it, so prime ✓
48 = 2 × 24, not prime
49 = 7 × 7, not prime
50 = 2 × 25, not prime
51 = 3 × 17, not prime
52 = 2 × 26, not prime
53: no prime up to √53 divides it, so prime ✓
54 = 2 × 27, not prime
55 = 5 × 11, not prime
56 = 2 × 28, not prime
57 = 3 × 19, not prime
58 = 2 × 29, not prime
59: no prime up to √59 divides it, so prime ✓
60 = 2 × 30, not prime
Write the primes.
41, 43, 47, 53, 59

Check. Count: 5 primes in the range. ✓

Watch out. Even numbers bigger than 2 are never prime, and numbers ending in 5 (bigger than 5) are never prime.

Worked example 2.2.4

Without working out the value, explain why 3 × 5 × 7 + 7 is not a prime number.

Method. Look for a common factor. If the whole expression has a factor other than 1 and itself, it cannot be prime.

Look at the two parts of the sum. There are two terms: 3 × 5 × 7 and 7. Both contain a 7.
3 × 5 × 7 + 7
Take the common factor out. Both terms are multiples of 7, so we can write the sum as 7 times a number.
3 × 5 × 7 + 7 = 7 × (3 × 5 + 1) = 7 × 16
Explain. The number has a factor of 7, which is neither 1 nor the number itself.
It is a multiple of 7 that is bigger than 7, so it is not prime.

Check. 3 × 5 × 7 + 7 = 105 + 7 = 112, and 112 = 7 × 16. ✓

Watch out. “Explain” needs a sentence naming the factor, not just the value.

Watch out

Do not forget that 1 is not prime and 2 is prime. Also, a number is not prime just because it is odd: 9, 15, 21, 25, 27 and 91 are odd but have other factors. And 91 looks prime but it is 7 × 13.

2.3

Prime factorisation

Edexcel 1MA1: N4
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Prime factorisationComing soon

Every whole number bigger than 1 can be built by multiplying primes. For example 12 = 2 × 2 × 3 and 30 = 2 × 3 × 5. Primes are the building blocks (the atoms) of the whole numbers. Writing a number as a product of primes is called finding its prime factorisation, or writing it as a product of prime factors.

One way is a factor tree. Split the number into any two factors. Then split each of those that is not prime. Keep going until every branch ends in a prime. Circle the primes as you reach them.

A factor tree for 60. 60 = 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5. The circled primes are 2, 3, 2, 5, so 60 = 2 × 2 × 3 × 5.

A different tree for 60, splitting off the smallest prime each time. The primes at the ends are the same: 2, 2, 3 and 5. The order does not matter.

The two trees look different but give the same primes. This is true for every number. It is called the unique factorisation theorem: every whole number above 1 can be written as a product of primes in only one way (apart from the order). That is why prime factors are so useful.

Another way is the division ladder: divide by the smallest prime that goes in (2, then 3, then 5, 7 ...), and keep dividing the answer until you reach 1. The numbers you divided by are the prime factors.

It is neater to write repeated primes with a power (this is called index form). 2 × 2 × 3 × 5 = 2² × 3 × 5.

Key ideas

A prime factorisation writes a number as a product of primes only.
Use a factor tree (split into any pair) or a division ladder (divide by the smallest prime each time).
Index form: 2 × 2 × 2 = 2³.
The unique factorisation theorem: there is only one prime factorisation of each number (ignoring order).
A number is a square if every power in its factorisation is even.

Worked example 2.3.1

Use a factor tree to write 60 as a product of prime factors. Give your answer in index form.

Key words. Writing a number as a product of prime factors means writing it as primes multiplied together. Index form writes repeated primes with a power: 2 × 2 × 2 = 2³. The unique factorisation theorem says every whole number above 1 has only one such product (apart from the order).

Method. Write 60 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

Split the number into any two factors (a factor pair). Choose two numbers that multiply to give it. Any pair will do: you will always reach the same primes in the end.
60 = 6 × 10
6 = 2 × 3
10 = 2 × 5
Keep splitting every number that is not prime. Circle each prime when you reach it. A prime cannot be split any more, so that branch stops. Stop when every branch ends in a circled prime.
Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
60 = 2 × 2 × 3 × 5
Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
60 = 2² × 3 × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 3 = 12
12 × 5 = 60
The result is 60, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Worked example 2.3.2

Write 360 as a product of prime factors. Give your answer in index form.

Method. Write 360 as a product of prime factors. Keep splitting into factors until every factor is a prime number.

Divide by the smallest prime that goes in. Keep going until you reach 1. We divide by the primes 2, 3, 5, 7 ... in order. Use each prime as many times as it goes in, then move to the next one.
360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
You can also show this as a factor tree. A tree is the same working drawn as branches. Prime ends are circled.
Collect the prime numbers you divided by. These (the circled numbers on the tree) are the prime factors.
360 = 2 × 2 × 2 × 3 × 3 × 5
Write repeated primes using powers (index form). For example 2 × 2 × 2 is three 2s, which is written 2³.
360 = 2³ × 3² × 5

Check. Multiply the primes back together:
2 × 2 = 4
4 × 2 = 8
8 × 3 = 24
24 × 3 = 72
72 × 5 = 360
The result is 360, the number we started with. ✓

Watch out. Every number at the end must be prime. 1 is not prime, and a number such as 9 or 15 is not finished because it can still be split.

Worked example 2.3.3

Work out the value of 2³ × 3² × 5.

Method. Evaluate each power first, then multiply.

Work out each power separately. A power means repeated multiplication of the base.
2³ = 2 × 2 × 2 = 8
3² = 3 × 3 = 9
5 = 5
Multiply the results together. The numbers in the product are multiplied.
8 × 9 = 72
72 × 5 = 360
Write the answer. The value of the number.
2³ × 3² × 5 = 360

Check. Prime factorise 360 again and you should get 2³ × 3² × 5. ✓

Watch out. 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.

Worked example 2.3.4

Show that 144 is a square number by using its prime factors, and write down its square root.

Method. A number is a square number only if every power in its prime factorisation is even.

Write the number as a product of primes (index form). Use a factor tree or repeated division by primes.
144 = 2⁴ × 3²
Look at the power of each prime. A square number is a number multiplied by itself, so every prime in it comes in pairs. That means every power must be even.
2: power 4 is even
3: power 2 is even
Decide. All the powers are even, so the number is a square. Halve every power to find the square root.
Yes: √144 = 2² × 3 = 12

Check. 12 × 12 = 144. ✓

Watch out. Look at every prime. One odd power is enough to stop it being a square.

Watch out

A factor tree is only finished when every branch ends in a prime. 9 and 15 are not primes, so they must be split again. Also write 2³ as 2 × 2 × 2 = 8, not 2 × 3 = 6.

2.4

Highest common factor and lowest common multiple

Edexcel 1MA1: N4
Grade 3-7

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Highest common factor and lowest common multipleComing soon

In Section 2.1 you met common factors and common multiples. Two special ones are used a lot.

The highest common factor (HCF) of two numbers is the biggest number that divides into both. We use it to cut things into the largest equal pieces, or to simplify fractions (Chapter 3).

The lowest common multiple (LCM) is the smallest number that is in both times tables. We use it when two things repeat and we want to know when they line up, or to find a common denominator.

Method 1: listing. For the HCF, list the factors of each number and find the biggest number in both lists. For the LCM, list the multiples of each number until you find the first one in both lists.

Method 2: prime factors and a Venn diagram. This method works for big numbers where lists would be long. Write each number as a product of primes. Draw two overlapping circles, one for each number. Put the primes that both numbers share in the middle, and the leftover primes in the outer parts.

12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Both share a 2 and a 3 (the middle). 12 has one more 2 and 18 has one more 3. HCF = 2 × 3 = 6. LCM = 2 × 3 × 2 × 3 = 36.

HCF = multiply the numbers in the middle. LCM = multiply all the numbers in the diagram (the middle ones only once). Why? The middle holds what both numbers have in common, which builds the biggest shared factor. The LCM has to contain everything that either number needs, and using the shared primes only once keeps it as small as possible.

60 = 2 × 2 × 3 × 5 and 84 = 2 × 2 × 3 × 7. The middle is 2, 2, 3 so HCF = 12. All the numbers multiplied: 2 × 2 × 3 × 5 × 7 = 420 = LCM.

Which one do I need? If the answer must be smaller than or equal to both numbers (cutting, sharing, grouping), it is the HCF. If it is bigger than or equal to both (meeting again, repeating, buying enough), it is the LCM.

Key ideas

HCF = the biggest number that divides into both. LCM = the smallest number in both times tables.
Listing works for small numbers. For larger numbers use prime factors and a Venn diagram.
Venn diagram: HCF = multiply the middle; LCM = multiply everything in the diagram.
HCF ≤ both numbers; LCM ≥ both numbers.
Check: HCF × LCM = the two numbers multiplied together.

Worked example 2.4.1

Find the HCF of 24 and 36 by listing factors.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. The highest common factor (HCF) is the biggest number that divides into both 24 and 36. Here we list the factors.

List the factors of 24. These are all the numbers that divide into it exactly.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
List the factors of 36. Do the same for the other number.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Find the common factors. These are the numbers that appear in both lists.
Common factors: 1, 2, 3, 4, 6, 12
Pick the highest one. The highest common factor is the biggest number in the common list.
HCF = 12

Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, and 2 and 3 have no common factor except 1, so no bigger number works. ✓

Watch out. HCF = highest factor (smaller than or equal to both numbers). Do not confuse it with the LCM, which is bigger than or equal to both.

Worked example 2.4.2

Find the LCM of 6 and 8 by listing multiples.

Method. The lowest common multiple (LCM) is the smallest number that is in both the 6 times table and the 8 times table. Here we list the multiples.

List the multiples of 6. Count up in 6s. Keep going until you have passed a number that is also in the other list.
Multiples of 6: 6, 12, 18, 24, ...
List the multiples of 8. Count up in 8s in the same way.
Multiples of 8: 8, 16, 24, ...
Find the first number that is in both lists. That is the lowest common multiple.
LCM = 24

Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3, both whole numbers. And no smaller number appears in both lists. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Worked example 2.4.3

Find the HCF and the LCM of 60 and 84.

Method. Find the HCF (biggest number dividing into both) and the LCM (smallest number in both times tables) of 60 and 84.

Write 60 and 84 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
60 = 2 × 2 × 3 × 5
84 = 2 × 2 × 3 × 7
Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
HCF: multiply the numbers in the middle (the overlap). The overlap holds the prime factors that both numbers share, so it builds the biggest number that divides into both.
HCF = 2 × 2 × 3 = 12
LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
LCM = 5 × 2 × 2 × 3 × 7 = 420

Check. HCF × LCM = 12 × 420 = 5040 and 60 × 84 = 5040. They are equal. ✓

Watch out. Check you give both answers, labelled HCF and LCM.

Worked example 2.4.4

Bus A leaves the station every 12 minutes. Bus B leaves every 18 minutes. Both leave at 9:00 am. At what time do they next leave together?

Method. The lowest common multiple (LCM) is the smallest number that is in both the 12 times table and the 18 times table. Here we use prime factors in a Venn diagram.

Decide what the question is asking. The buses leave together again after a number of minutes that is in both the 12 times table and the 18 times table. We want the first time, so we need the lowest common multiple.
Find the lowest common multiple of 12 and 18.
Write 12 and 18 as products of prime factors. Use a factor tree or repeated division by primes. Prime factors are the building blocks of each number.
12 = 2 × 2 × 3
18 = 2 × 3 × 3
Put the primes in a Venn diagram. A prime that is in both numbers goes in the middle (where the circles overlap). A prime that belongs to only one number goes in that circle only. If a prime is repeated, e.g. 2 × 2, put as many copies in the middle as both numbers have.
LCM: multiply all the numbers in the diagram (the overlap counted once). Everything in the diagram is needed to build a number that both can divide into, and the shared primes are used only once to keep it as small as possible.
LCM = 2 × 2 × 3 × 3 = 36
Answer the question in words. Always finish by saying what the number means.
They leave together again after 36 minutes, which is 9:36 am.

Check. HCF × LCM = 6 × 36 = 216 and 12 × 18 = 216. They are equal. ✓

Watch out. LCM = lowest multiple (bigger than or equal to both numbers). Do not just multiply the two numbers: that gives a common multiple, but often not the lowest.

Watch out

Do not mix the two up: HCF is a factor (small), LCM is a multiple (big). And the LCM is not always the two numbers multiplied together: LCM of 6 and 8 is 24, not 48.

2.5

Systematic listing and the product rule

Edexcel 1MA1: N5
Grade 2-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Systematic listing and the product ruleComing soon

Many questions ask “how many different ways ...?” How many outfits can you make? How many different two-digit numbers? How many ways can you pay 30p? To answer, you need to find all the possibilities without missing any and without counting any twice.

The trick is to be systematic. A systematic list uses a plan:

Decide the order of the choices (for example, T-shirt first, then trousers).
Fix the first choice and list every way to make the other choices.
Move to the next first choice and repeat.
Count at the end.

For numbers, list them in order of size: start with the smallest first digit, then the smallest second digit, and so on.

A tree diagram shows a systematic list. Each T-shirt colour (Red, Blue, Green) is joined to each choice of bottoms (Jeans, Shorts). There are 3 × 2 = 6 ends, so 6 outfits.

Useful ideas: for totals (such as two dice adding to 7) go through the first number in order and work out what the second must be. For coin questions start with as many of the biggest coin as you can, then use one fewer each time.

Counting without listing. Look at the tree: 3 colours, each with 2 choices, gives 3 × 2 = 6. This is a shortcut: if there are m ways to do one thing and n ways to do another, you can multiply to count the total.

Key ideas

Be systematic: choose an order, fix the first choice, list the rest, then move on.
List numbers in order of size so nothing is missed.
Count only when the list is finished.
A quick check: with m choices then n choices there are m × n combinations.
Product rule (Higher): multiply the number of choices at each stage.
Fill any restricted position first.

Worked example 2.5.1

A cafe sells 3 T-shirt colours (Red, Blue, Green) and 2 types of bottoms (Jeans, Shorts). List all the outfits and say how many there are.

Key words. A combination here is one choice from each group. A systematic list is a list written in a planned order.

Method. Make a systematic list: fix the first choice, list every combination for it, then move on to the next first choice.

Decide on an order and stick to it. A systematic list means you take the choices in a fixed order, so you cannot miss one or list one twice. Here we fix the choice of colour first.
3 colour, 2 bottoms
Take the first choice of colour and pair it with every possible choice of the rest. Keep the first one fixed while the others change one at a time.
Red: Jeans, Shorts
Repeat for each of the other choices of colour. Work through the remaining choices in order.
Blue: Jeans, Shorts
Green: Jeans, Shorts
Count the combinations. Count every item in your list.
Total = 6

Check. Multiply the number of choices: 3 × 2 = 6. This is the product rule and it agrees with the list. ✓

Watch out. Without a system it is easy to repeat a combination or miss one. Always write your list in an organised order.

Worked example 2.5.2

Using each of the digits 2, 5 and 7 once, write down all the three-digit numbers that can be made.

Method. Write the numbers in order, one first digit at a time.

Read the rules. Each digit can be used only once.
Digits: 2, 5, 7 Length: 3 digits
List the numbers in order of size, one first digit at a time. Fixing the first digit, then the next, gives a list in which nothing can be missed.
Starting with 2: 257, 275
Starting with 5: 527, 572
Starting with 7: 725, 752
Count them. Count every number in the list.
Total = 6

Check. The list is in increasing order with no repeats, so every number is counted once. ✓

Watch out. Check the rules: whether digits can repeat and any condition such as “even” or “bigger than 300”.

Worked example 2.5.3

Two fair dice are thrown. Their scores are added. List all the ways to get a total of 8.

Method. Make a systematic list: take the first number in order and find the second number that gives 8.

Go through the first number in order. For each first number, work out what the second number would have to be.
First = 1: no second number works
First = 2: second = 6 (2 + 6 = 8) ✓
First = 3: second = 5 (3 + 5 = 8) ✓
First = 4: second = 4 (4 + 4 = 8) ✓
First = 5: second = 3 (5 + 3 = 8) ✓
First = 6: second = 2 (6 + 2 = 8) ✓
Write the pairs. List them as (first, second).
(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
Count them. Count the pairs.
5 pairs

Check. Each pair really gives 8: 2 + 6 = 8, 3 + 5 = 8, 4 + 4 = 8, 5 + 3 = 8, 6 + 2 = 8. ✓

Watch out. (2, 5) and (5, 2) are different outcomes when the two numbers come from different dice.

Worked example 2.5.4

Write down all the ways of making 30p using 20p, 10p and 5p coins.

Method. Make a systematic list for making 30p. Fix the number of the biggest coin, then list the ways to make the rest.

Start with the most of the biggest coin you can use. Working from the largest coin down means every possibility is reached in turn.
Coins: 20p, 10p, 5p
Reduce the number of the biggest coin one at a time and fill the rest with smaller coins. For each choice, find all the ways to make up the remainder.
1 × 20p + 1 × 10p = 30p
1 × 20p + 2 × 5p = 30p
3 × 10p = 30p
2 × 10p + 2 × 5p = 30p
1 × 10p + 4 × 5p = 30p
6 × 5p = 30p
Count the ways. Count the lines.
6 ways

Check. Each line adds up to 30p, and no two lines are the same. ✓

Watch out. Be organised: write the number of each coin in the same order every time.

Watch out

Do not just write down numbers at random. A messy list almost always misses some. Also check the rules: can digits be repeated? Does the order matter (12 and 21 are different numbers)? And with the product rule, make sure you multiply the numbers of choices. Do not add them.

2.6

Squares, cubes, roots and powers

Edexcel 1MA1: N3, N6
Grade 1-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Squares, cubes, roots and powersComing soon

To square a number, multiply it by itself: 5 squared is 5 × 5 = 25, written 5². It is called squaring because 5 × 5 is the area of a square with side 5. The numbers 1, 4, 9, 16, 25 ... are the square numbers.

Square numbers: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25. Each is the number of small squares in a square with that side.

To cube a number, multiply three copies: 3³ = 3 × 3 × 3 = 27. It is the volume of a cube with edge 3. The numbers 1, 8, 27, 64, 125 ... are the cube numbers.

A cube with edge 3 cm is made from 3 × 3 × 3 = 27 small cubes, so its volume is 27 cm³.

Roots undo powers. The square root of 49, written √49, is the number that squares to give 49. Since 7 × 7 = 49, √49 = 7. The cube root of 64 is 4, because 4 × 4 × 4 = 64. Roots of other powers work the same way: the fourth root of 81 is 3 because 3⁴ = 81. The √ sign gives the positive root, but the equation x² = 49 has two solutions, 7 and −7, because (−7) × (−7) = 49 too.

Learn these. Squares up to 15²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Cubes: 1, 8, 27, 64, 125 and 10³ = 1000.

Powers of 2, 3, 4 and 5. 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 | 3, 9, 27, 81, 243, 729 | 4, 16, 64, 256, 1024 | 5, 25, 125, 625, 3125. Each is the one before, multiplied by the base. Powers of 10 are easy: 10² = 100, 10³ = 1000, 10⁴ = 10 000.

The powers of 2 grow quickly: each bar is double the one before.

Powers and BIDMAS. Powers and roots (Indices) come after Brackets and before multiplying or adding. So 3 + 2² = 3 + 4 = 7, but (3 + 2)² = 5² = 25. The root sign works like a bracket: √(9 + 16) means work out 9 + 16 = 25 first, then take the root to get 5. And note that −3² = −9 but (−3)² = 9.

Squares of decimals and fractions: 0.3² = 0.09, and (2/3)² = 4/9. Squaring a number between 0 and 1 gives something smaller.

Key ideas

n² = n × n and n³ = n × n × n.
√ (square root) and the cube root undo squaring and cubing.
Learn the squares to 15², the cubes 1, 8, 27, 64, 125, 1000 and the powers of 2, 3, 4, 5.
BIDMAS: powers and roots are done before multiplying and adding. A root sign acts like a bracket.
−3² = −9 but (−3)² = 9.

Worked example 2.6.1

Work out 13² without a calculator.

Key words. To square a number, multiply it by itself (5² = 5 × 5 = 25). To cube a number, multiply three of it (5³ = 5 × 5 × 5 = 125). A square root (√) undoes squaring: √25 = 5. A cube root undoes cubing: the cube root of 125 is 5.

Method. To square a number, multiply it by itself.

Write it as a multiplication. Squaring means multiplying the number by itself.
13² = 13 × 13
Split one of the numbers into tens and units and multiply each part. This is long multiplication done in two easy steps: 13 × 10 and 13 × 3.
13 × 10 = 130
13 × 3 = 39
Add the two parts. Add the results of the two multiplications.
130 + 39 = 169

Check. The last digit of 13² comes from 3 × 3 = 9, so the answer should end in 9. ✓

Watch out. 13² means 13 × 13, not 13 × 2.

Worked example 2.6.2

Work out 6³.

Method. To cube a number, multiply three copies of it together.

Write it as a multiplication. Cubing means multiplying three copies of the number.
6³ = 6 × 6 × 6
Do the first two numbers first. Multiply in stages so that each stage is easy.
6 × 6 = 36
Multiply by the third number. Now multiply the answer by the last copy.
36 × 6 = 216

Check. Go the other way: the cube root of 216 should be 6, and 6 × 6 × 6 = 216. ✓

Watch out. 6³ means 6 × 6 × 6, not 6 × 3.

Worked example 2.6.3

Work out √169.

Method. A square root is the number that, multiplied by itself, gives the number inside the root.

Think: which number multiplied by itself gives 169? A square root undoes a square.
We want ? × ? = 169
Look at the squares you know that are close to the number. Use your list of square numbers to home in on the answer.
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
Pick the one that matches. Only one square equals the number.
13 × 13 = 169, so √169 = 13

Check. Square the answer: 13² = 169. ✓

Watch out. √169 means the positive number 13. (The equation x² = 169 has two answers, 13 and −13, but the √ sign gives only the positive one.)

Worked example 2.6.4

Write down the cube root of 343.

Method. A cube root is the number that, multiplied by itself three times, gives the number.

Think: which number multiplied by itself three times gives 343? A cube root undoes a cube.
We want ? × ? × ? = 343
Look at the cubes you know. Use your list of cube numbers.
6³ = 216
7³ = 343
8³ = 512
9³ = 729
Pick the one that matches. Only one cube equals the number.
7 × 7 × 7 = 343, so the cube root of 343 is 7

Check. Cube the answer: 7³ = 343. ✓

Watch out. The cube root of 343 is 7, not 343 ÷ 3.

Worked example 2.6.5

Write 625 as a power of 5.

Key words. A power of a number is that number multiplied by itself a certain number of times.

Method. Keep multiplying by 5 until you reach 625, and count how many 5s you used.

Start with the base and multiply by it again and again. Each line is one more power of the base.
5¹ = 5
5² = 25
5³ = 125
5⁴ = 625
Read off the power. 625 appears after 4 multiplications, so the power is 4.
625 = 5⁴
Check by counting. The number of 5s multiplied is the index.
5 × 5 × 5 × 5 = 625

Check. 5⁴ = 625. ✓

Worked example 2.6.6

Work out √(5² + 12²).

Key words. BIDMAS is the order for working out: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Method. Use BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.

Write the calculation and decide what must be done first. Spot the brackets and the powers or roots.
√(5² + 12²)
Work out the powers inside the root. The root sign acts like a bracket: do the working underneath it first.
5² = 25
12² = 144
Add inside the root. Add the two squares.
25 + 144 = 169
Take the square root. Now find the number that squares to give the total.
√169 = 13

Check. Work through the order again step by step, or use a calculator with brackets, to confirm 13. ✓

Watch out. You cannot split it: √(5² + 12²) is not 5 + 12 = 17.

Watch out

Squaring is not doubling: 6² = 36, not 12. And a cube root is not “divide by 3”: the cube root of 27 is 3, not 9. Finally, be careful with negatives: −5² = −25 but (−5)² = 25.

2.7

Powers and the laws of indices

Edexcel 1MA1: N7
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Powers and the laws of indicesComing soon

A power (or index) tells you how many times to multiply a number by itself. In 2⁵ the base is 2 and the index is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The plural of index is indices. Powers let us write long multiplications briefly. The laws of indices are shortcuts for working with powers of the same base.

Law 1: multiplying. What is 2³ × 2⁴? Write out the copies of 2:

2³ × 2⁴ has three 2s and four 2s multiplied together: seven 2s in all, so it is 2⁷.

So 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128. When you multiply powers of the same base, add the indices.

Law 2: dividing. 2⁶ ÷ 2² = (2 × 2 × 2 × 2 × 2 × 2) ÷ (2 × 2). Two 2s cancel top and bottom, leaving four 2s. So 2⁶ ÷ 2² = 2⁶⁻² = 2⁴ = 16. When you divide powers of the same base, subtract the indices.

Law 3: a power of a power. (2³)² = 2³ × 2³ = 2³⁺³ = 2⁶. That is 2 lots of 3 copies, which is 2 × 3 = 6 copies. When you raise a power to a power, multiply the indices.

Key ideas

Multiply: a^m × aⁿ = a^m+n (add the indices).
Divide: a^m ÷ aⁿ = a^m−n (subtract the indices).
Power of a power: (a^m)ⁿ = a^mn (multiply the indices).
The laws only work when the bases are the same. If they are different, try to rewrite them as powers of the same base (for example 4 = 2²).
Do powers before multiplying (BIDMAS).

Worked example 2.7.1

Write 2³ × 2⁴ as a single power of 2, and work out its value.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To multiply powers of the same base, add the indices: a^m × aⁿ = a^m+n.

Write each power out in full. 2³ is 3 copies of 2 multiplied, and 2⁴ is 4 copies.
2³ = 2 × 2 × 2
2⁴ = 2 × 2 × 2 × 2
Join them into one multiplication and count the copies of 2. Multiplying the two powers puts all the copies together: 3 + 4 copies.
3 + 4 = 7 copies of 2
Write the answer as a single power. The number of copies is the new index. This is the rule: add the indices when you multiply powers of the same base.
2³ × 2⁴ = 2⁷
Work out the value. Multiply the copies out.
2⁷ = 128

Check. Work out each power: 8 × 16 = 128, which is 2⁷. ✓

Watch out. Add the indices, do not multiply them: 2³ × 2⁴ = 2⁷, not 2¹². And the base stays the same.

Worked example 2.7.2

Write 5⁶ ÷ 5² as a single power of 5, and work out its value.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To divide powers of the same base, subtract the indices: a^m ÷ aⁿ = a^m−n.

Write the division as a fraction with each power written in full. A fraction line means divide, and it lets us cancel.
5⁶ ÷ 5² = 5 × 5 × 5 × 5 × 5 × 55 × 5
Cancel the copies of 5 that appear on the top and the bottom. Each 5 on the bottom cancels with one on the top, because 5 ÷ 5 = 1. We cancel 2 of them.
6 − 2 = 4 copies of 5 left on the top
Write the answer as a single power. The copies left on the top are the new index. This is the rule: subtract the indices when you divide powers of the same base.
5⁶ ÷ 5² = 5⁴
Work out the value. Multiply the copies out.
5⁴ = 625

Check. 15,625 ÷ 25 = 625. ✓

Watch out. Subtract the indices, do not divide them. The base stays 5.

Worked example 2.7.3

Write (3²)³ as a single power of 3, and work out its value.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. To raise a power to a power, multiply the indices: (a^m)ⁿ = a^mn.

Write the outer power as repeated multiplication of the bracket. (3²)³ means 3 copies of 3² multiplied together.
(3²)³ = 3² × 3² × 3²
Each bracket holds 2 copies of 3. Count the copies altogether. 3 brackets with 2 copies in each.
3 × 2 = 6 copies of 3
Write the answer as a single power. This is the rule: multiply the indices when you raise a power to a power.
(3²)³ = 3⁶
Work out the value. Multiply the copies out.
3⁶ = 729

Check. 9³ = 729 and 3⁶ = 729. ✓

Watch out. (3²)³ multiplies the indices (2 × 3 = 6). Do not add them: 2 + 3 = 5 is the rule for multiplying two separate powers.

Worked example 2.7.4

Write (2⁵ × 2⁴) ÷ 2³ as a single power of 2.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index laws one at a time, in BIDMAS order: add indices to multiply, subtract to divide, multiply for a power of a power.

Write the expression. Look for brackets and decide the order.
(2⁵ × 2⁴) ÷ 2³
Do the multiplication on the top first. Multiplying powers of the same base: add the indices.
2⁵ × 2⁴ = 2⁹ (5 + 4 = 9)
Now divide. Dividing powers of the same base: subtract the indices.
2⁹ ÷ 2³ = 2⁶ (9 − 3 = 6)

Check. Evaluate the original with numbers: 32 × 16 ÷ 8 = 64. ✓

Watch out. Only combine indices when the bases are the same. 2³ × 3² cannot be written as one power.

Worked example 2.7.5

Find the value of n when 3ⁿ × 3⁴ = 3⁹.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the index law to write the left side as a single power, then make the indices equal.

Write the equation. We want the value of n.
3ⁿ × 3⁴ = 3⁹
Use the multiplication law on the left. Add the indices.
3ⁿ × 3⁴ = 3^{n + 4}
Make the indices equal. The bases are the same, so the indices must match.
n + 4 = 9
Solve. Subtract 4 from both sides.
n = 9 − 4 = 5

Check. 3⁵ × 3⁴ = 3⁹ = 3⁹. ✓

Watch out. Use the law in the correct direction: add for multiply, subtract for divide, multiply for a power of a power.

Watch out

You add the indices when multiplying, not multiply them: 2³ × 2⁴ = 2⁷, not 2¹². And the base does not change: 2³ × 2⁴ is not 4⁷. The laws do not work for adding: 2³ + 2⁴ = 8 + 16 = 24, which is not 2⁷.

2.8

Zero and negative indices

Edexcel 1MA1: N7
Grade 4-7

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Zero and negative indicesComing soon

What happens when the index is 0 or negative? Look at the pattern of powers of 2. Going down the list, each power is the one above divided by 2.

Power	2³	2²	2¹	2⁰	2⁻¹	2⁻²	2⁻³
Value	8	4	2	1	12	14	18

The pattern gives two rules.

Zero index. 2⁰ = 1. In fact, any number (except 0) to the power 0 is 1. You can see why from the division law: 2³ ÷ 2³ = 2³⁻³ = 2⁰, but anything divided by itself is 1.

Negative index. A negative index means “one over”: 2⁻³ = 1 ÷ 2³ = 18. In general a⁻ⁿ = 1aⁿ. It is the reciprocal of the positive power. A negative index does not make the number negative: 2⁻³ is a small positive fraction, not −8 and not −6.

The index laws still work with negative indices. For example 5² ÷ 5⁴ = 5²⁻⁴ = 5⁻² = 125.

Key ideas

a⁰ = 1 for any a that is not 0.
a⁻ⁿ = 1 ÷ aⁿ: the reciprocal of aⁿ.
A negative index gives a fraction, never a negative number (if the base is positive).
The index laws work for all integer indices: add to multiply, subtract to divide.
(Higher) (a/b)⁻ⁿ = (b/a)ⁿ.

Worked example 2.8.1

Write down the value of 9⁰.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

Write down the pattern of powers of 9, going down. Each time the index goes down by 1 we divide by 9.
9³ = 729
9² = 81
9 = 9
Continue the pattern for index 0. Dividing 9 by 9 gives 1. This also follows from the division law: 9³ ÷ 9³ = 9⁰, and anything divided by itself is 1.
9⁰ = 9 ÷ 9 = 1
Write the answer. The same works for every base, so the answer does not depend on the base.
9⁰ = 1

Check. 9⁴ ÷ 9⁴ = 9⁰ and also = 1. ✓

Watch out. 9⁰ is 1, not 0 and not 9.

Worked example 2.8.2

Work out 2⁻³. Give your answer as a fraction.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 2.
2³ = 8
2² = 4
2 = 2
2⁰ = 1
2⁻¹ = 12
2⁻² = 14
2⁻³ = 18
Spot the rule. A negative index means “one over”: 2⁻³ is the reciprocal of 2³.
2⁻³ = 12³
Work out the value of the bottom. Evaluate the positive power.
2³ = 8, so 2⁻³ = 18

Check. 2⁻³ × 2³ should be 2⁰ = 1: 18 × 8 = 1. ✓

Watch out. 2⁻³ is 18, not −8. The minus sign in the index changes it to a reciprocal.

Worked example 2.8.3

Work out 10⁻² as a fraction and as a decimal.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 10.
10³ = 1000
10² = 100
10 = 10
10⁰ = 1
10⁻¹ = 110
10⁻² = 1100
Spot the rule. A negative index means “one over”: 10⁻² is the reciprocal of 10².
10⁻² = 110²
Work out the value of the bottom. Evaluate the positive power.
10² = 100, so 10⁻² = 1100
Write it as a decimal as well. Divide 1 by 100.
1 ÷ 100 = 0.01

Check. 10⁻² × 10² should be 10⁰ = 1: 1100 × 100 = 1. ✓

Watch out. 10⁻² is 1100, not −100. The minus sign in the index changes it to a reciprocal.

Worked example 2.8.4

Work out 5² ÷ 5⁴.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

Check the bases are the same. Both powers have base 5, so we can use the index law.
5² ÷ 5⁴
Subtract the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
2 − 4 = −2
5² ÷ 5⁴ = 5⁻²
Write the negative index as a fraction. A negative index means 1 divided by the positive power.
5⁻² = 15² = 125

Check. Work out each power as a number first, then multiply or divide: 25 ÷ 625 = 125. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Worked example 2.8.5

Work out 2³ ÷ 2⁻².

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

Check the bases are the same. Both powers have base 2, so we can use the index law.
2³ ÷ 2⁻²
Subtract the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
3 − (−2) = 5
2³ ÷ 2⁻² = 2⁵
Work out the value. Multiply the copies out.
2⁵ = 32

Check. Work out each power as a number first, then multiply or divide: 8 ÷ 14 = 32. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Watch out

A negative index does not make the answer negative. 3⁻² = 19, not −9 and not −6. Also 5⁰ = 1, not 0 and not 5.

Exercise 2H Zero and negative indices

Non-calculator

FWrite down the value of each of these.

(a)5⁰

(b)8⁰

(d)12⁰

(e)100⁰

(f)3⁰

Done

FWork out each of these. Give your answer as a fraction.

(a)2⁻²

(b)3⁻²

(d)10⁻³

(e)4⁻²

(f)2⁻⁴

Done

FWrite each number as a fraction and as a decimal.

(a)2⁻³

(b)5⁻²

(d)4⁻¹

(e)10⁻¹

(f)2⁻²

Done

FWrite each fraction as a power of the base given, using a negative index.

(a)19 (base 3)

(b)18 (base 2)

(d)1100 (base 10)

(e)132 (base 2)

(f)181 (base 3)

Done

FWork out each of these.

(a)2⁵ × 2⁻²

(b)3⁻¹ × 3⁴

(d)10³ × 10⁻⁵

(e)2⁻³ ÷ 2⁻⁵

(f)4⁻² × 4²

Done

PSPut these numbers in order, smallest first: 2⁻³, 2⁰, 2³, 2⁻¹, 14.

Done

RExplain why 2⁰ = 1. Use the index law for dividing.

Done

RDecide whether each statement is true or false. Give a reason.

(a)3⁻² = −9

(b)2⁻¹ = 0.5

(d)4⁻² = 116

Done

RWhich is bigger, 2⁻³ or 3⁻²? Show how you know.

Done

2.9

Estimating roots and powers, and using a calculator

Edexcel 1MA1: N6, N7
Grade 3-7

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Estimating roots and powers, and using a calculatorComing soon

Most numbers are not perfect squares. For example 30 is not a square number, so √30 is not a whole number. But we can still say roughly how big it is. Look at the squares either side of 30: 5² = 25 and 6² = 36. Since 25 < 30 < 36, the square root of 30 is between 5 and 6.

√30 lies between 5 and 6. It is closer to 5.5 than to 5 or 6, at about 5.48.

Which whole number is it nearer to? The halfway number is 5.5, and 5.5² = 30.25. Because 30 is a little less than 30.25, √30 is a little less than 5.5, so it is nearer to 5.

Estimating to 1 decimal place (trial and improvement). Square numbers with one decimal place between 5 and 6: 5.4² = 29.16 (too small), 5.5² = 30.25 (too big). So √30 is between 5.4 and 5.5. To choose, test the halfway value 5.45: 5.45² = 29.7025, which is less than 30, so √30 is above 5.45 and rounds to 5.5.

Cube roots work the same way with cubes: 3³ = 27 and 4³ = 64, so the cube root of 50 is between 3 and 4.

Estimating powers. Round the base to an easy number first. 3.9² is close to 4² = 16. Because 3.9 is slightly less than 4, the true answer is slightly less than 16 (it is 15.21).

Using a calculator. Use the x² and x³ keys for squares and cubes, the power key (x^□ or ^) for other powers, and the √ key for roots (on many calculators you press √ first, then the number, then =). Use brackets for negative numbers and for working under a root: √(6.4² − 3.2²) needs brackets round the whole of the working under the root. Always estimate first, to catch key-pressing mistakes.

Key ideas

Roots of non-square numbers lie between two whole numbers: find the squares (or cubes) either side.
To decide which is nearer, test the halfway number.
To get 1 d.p., square the one-decimal-place numbers (trial and improvement) and test the halfway value.
Estimate a power by rounding the base first.
On a calculator use the power and root keys, use brackets, and check with an estimate.

Worked example 2.9.1

Write down the two whole numbers between which √30 lies. Which is it nearer to?

Key words. To estimate is to find an answer that is close, without finding it exactly. A number that is not a perfect square or cube has a root that is a decimal.

Method. Find the two whole numbers whose squares are either side of 30.

Find the square numbers just below and just above 30. Use the square numbers you know.
5² = 25
6² = 36
25 < 30 < 36
Take the square root of each part of the inequality. Taking a square root keeps the order, so the root of 30 is between 5 and 6.
5 < √30 < 6
Decide which whole number is nearer. Halfway between 5 and 6 is 5.5, and 5.5² = 30.25. If 30 is below 30.25 the root is below halfway.
30 is less than 30.25, so √30 is nearer to 5.

Check. Use a calculator: √30 = 5.477..., which is between 5 and 6 and nearer to 5. ✓

Watch out. √30 is not a whole number here, so do not give an exact whole-number answer.

Worked example 2.9.2

Find √30 to 1 decimal place without a calculator.

Key words. Trial and improvement means trying a value, seeing if it is too big or too small, and adjusting it. 1 decimal place (1 d.p.) means one digit after the decimal point.

Method. Use trial and improvement: square one-decimal-place numbers until you have trapped 30 between two of them.

Find the whole numbers either side. Use the square numbers you know.
5² = 25 and 6² = 36, so 5 < √30 < 6
Try one-decimal-place numbers between them, using the position of 30 as a guide. 30 is closer to 25 than to 36, which tells us where to start. Work out the square of each trial.
5.4² = 29.16
5.5² = 30.25
5.6² = 31.36
Find the two one-decimal-place numbers that 30 lies between. We want one trial answer that is too small and the next one that is too big.
5.4² = 29.16 < 30 < 30.25 = 5.5²
Test the halfway value to decide which one-decimal-place number is nearer. Halfway between 5.4 and 5.5 is 5.45. If 30 is bigger than 5.45² the root is bigger than 5.45 and rounds up.
5.45² = 29.7025
30 is more than 29.7025, so √30 is above 5.45.
Write the answer to 1 decimal place.
√30 ≈ 5.5

Check. A calculator gives √30 = 5.4772..., which rounds to 5.5 to 1 decimal place. ✓

Watch out. To round correctly, always test the halfway value. Do not just pick the trial answer that looks closer.

Worked example 2.9.3

Estimate the value of 3.9².

Key words. ≈ means “approximately equal to”.

Method. To estimate a power, round the base first (to the nearest whole number or 1 significant figure), then work out the power.

Round 3.9 to a whole number. A whole number is easier to square or cube in your head.
3.9 ≈ 4
Work out the power of the rounded number. Use the rounded value in place of the original.
4² = 16
Say it is an estimate. Use the ≈ sign, which means “is approximately equal to”.
3.9² ≈ 16

Check. The exact value is 15.21, which is close to 16. Because 3.9 is smaller than 4, the exact answer is smaller than the estimate. ✓

Watch out. Round the base before squaring or cubing, not the answer.

Worked example 2.9.4

Use a calculator to work out 1.7⁴. Give your answer to 2 decimal places.

Key words. d.p. means decimal places and s.f. means significant figures.

Method. Use the power key on the calculator, then round as asked.

Enter the number and use the power key. Your calculator has a key that raises a number to a power. Enter the base first.
Press 1.7, then the power key, then 4, then =.
Read the display. The calculator shows the full answer.
1.7⁴ = 8.3521
Round to 2 decimal places. Look at the next digit: 5 or more rounds up.
1.7⁴ = 8.35 to 2 decimal places

Check. Estimate: round 1.7 to 2 and work out 2⁴ = 16. This is close to 8.35. ✓

Watch out. Use brackets for a negative base: (−3)² is 9 but −3² is −9.

Worked example 2.9.5

Use a calculator to work out √77. Give your answer to 2 decimal places.

Key words. d.p. means decimal places.

Method. Use the root key on the calculator, then round as asked.

Use the √ key. Your calculator finds roots with a special key. On many calculators you press the root key first, then enter the number, then =.
√77 = 8.774964...
Round to 2 decimal places. Look at the digit after the last one you keep: 5 or more rounds up.
√77 = 8.77 to 2 d.p.
Check that the answer is sensible. Find the square numbers either side of 77: the root must be between the two whole numbers.
8² = 64 < 77 < 81 = 9², so the root is between 8 and 9. ✓

Check. Check by raising to the power: 8.77² = 76.9129, which is close to 77. ✓

Watch out. A calculator answer is usually a long decimal. Do not write the whole display: round as asked.

Watch out

Do not give a calculator answer with all its digits unless the question asks for it: round as asked. And a root that is not a whole number cannot be found exactly as a decimal: 5.48 is only an approximation of √30, so we use the sign ≈.

Exercise 2J Powers and roots with a calculator

Calculator allowed

FUse a calculator to work out each power.

(a)1.7⁴ (2 d.p.)

(b)2.3³ (1 d.p.)

(d)12.5³ (3 s.f.)

(e)3.2⁵ (1 d.p.)

(f)1.05¹⁰ (3 d.p.)

Done

FUse a calculator to work out each root.

(a)√77 (2 d.p.)

(b)√150 (2 d.p.)

(c)the cube root of 250 (1 d.p.)

(d)√2 (3 d.p.)

(e)the cube root of 90 (2 d.p.)

(f)√1000 (1 d.p.)

Done

FUse a calculator to work out each of these. Write down the estimate you used to check.

(a)(2.5 + 1.3)² (1 d.p.)

(b)√(6.4² − 3.2²) (2 d.p.)

(c)the cube root of (2.5³ + 1.5³) (2 d.p.)

(d)1.2³ × 4.5 (2 d.p.)

Done

PSA square has an area of 250 cm². Work out the length of one side. Give your answer to 1 decimal place.

Done

PSA cube has a volume of 500 cm³. Work out the length of one edge. Give your answer to 1 decimal place.

Done

PSA single cell splits into two every 20 minutes. After 24 splits (8 hours), the number of cells is 2²⁴. Work out this number.

Done

RZoe uses her calculator to work out −4² and gets −16. Max works out (−4)² and gets 16. Zoe says: “We did the same sum, so one of us is wrong.” Explain what is happening.

Done

RFind the smallest whole number n for which 1.5ⁿ is bigger than 1000.

Done

Chapter summary

Skill	Method
Multiples and factors	Multiples: the times table of the number. Factors: find pairs that multiply to the number. Use divisibility tests (2, 3, 4, 5, 6, 8, 9, 10).
Prime numbers	Exactly two factors, 1 and itself. 1 is not prime; 2 is the only even prime. To test: divide by primes up to the square root.
Prime factorisation	Factor tree or division ladder, then index form: 360 = 2³ × 3² × 5. Every number has only one factorisation.
HCF and LCM	List, or write as primes and use a Venn diagram: HCF = multiply the middle; LCM = multiply everything. HCF × LCM = product of the two numbers.
Systematic listing	Fix the first choice, list the rest in order, then count. Numbers in order of size.
Product rule (Higher)	Multiply the number of choices at each stage. Fill any restricted position first.
Squares, cubes, roots	Learn squares to 15², cubes 1, 8, 27, 64, 125, 1000, and powers of 2, 3, 4, 5. √ and the cube root undo the powers.
Laws of indices	Same base: add the indices to multiply, subtract to divide, multiply for a power of a power.
Zero and negative indices	a⁰ = 1 and a⁻ⁿ = 1 ÷ aⁿ. (Higher) (a/b)⁻ⁿ = (b/a)ⁿ.
Estimating roots and powers	Find the squares (cubes) either side. For 1 d.p., test one-decimal-place numbers and the halfway value. Round the base to estimate a power.

Can you…?

I can list the multiples and factors of a number and use divisibility tests.
I can say whether a number is prime and list the primes up to 50.
I can write a number as a product of prime factors in index form.
I can find the HCF and LCM by listing and with a Venn diagram, and use them to solve problems.
I can list outcomes systematically and count them.
(Higher) I can use the product rule for counting.
I know the squares to 15², the common cubes and the powers of 2, 3, 4 and 5, and I can find roots.
I can use the laws of indices and the rules for zero and negative indices.
I can estimate square and cube roots and use a calculator for powers and roots.
(Higher) I can find the number of factors, make a number a perfect square, and work with fractional bases and negative indices.

Key terms

Multiple: A number in the times table of another number, e.g. 12 is a multiple of 3.
Factor: A whole number that divides exactly into another, e.g. 4 is a factor of 12.
Common factor / common multiple: A factor (or multiple) shared by two or more numbers.
Prime number: A number with exactly two factors, 1 and itself.
Composite number: A number with more than two factors.
Prime factorisation: Writing a number as a product of prime numbers only.
Index form: Writing repeated multiplication as a power, e.g. 2 × 2 × 2 = 2³.
Unique factorisation theorem: Every whole number above 1 has exactly one prime factorisation, apart from the order.
HCF: Highest common factor: the biggest number that divides into two or more numbers.
LCM: Lowest common multiple: the smallest number in all of the times tables.
Venn diagram: A diagram of overlapping circles; here it shows the prime factors that two numbers share.
Systematic listing: Listing possibilities in a planned order so that none is missed or repeated.
Product rule for counting: If there are m ways of doing one task and n ways of doing another, there are m × n ways of doing both.
Square number: The result of multiplying a whole number by itself: 1, 4, 9, 16 ...
Cube number: The result of multiplying a whole number by itself twice more: 1, 8, 27, 64 ...
Square root / cube root: The number that, squared (cubed), gives the original number.
Index (power, exponent): The small number that says how many copies of the base are multiplied.
Base: The number that is being multiplied by itself in a power.
Reciprocal: 1 divided by a number. The reciprocal of 4 is 14.
Trial and improvement: Trying values, checking whether they are too big or too small, and adjusting.

Total 31 marks. Allow about 31 minutes.

← Chapter 1: Number basics Chapter 3: Fractions →

GCSE Maths Factors, Multiples, Primes and Powers (Foundation): worked examples and questions with answers

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Multiples and factors

Divisibility tests

Key ideas

Watch out

Exercise 2A Multiples and factors

Prime numbers

Key ideas

Watch out

Exercise 2B Prime numbers

Prime factorisation

Key ideas

Watch out

Exercise 2C Prime factorisation

Highest common factor and lowest common multiple

Key ideas

Watch out

Exercise 2D HCF and LCM

Systematic listing and the product rule

Key ideas

Watch out

Exercise 2E Systematic listing and the product rule

Squares, cubes, roots and powers

Key ideas

Watch out

Exercise 2F Squares, cubes and roots

Powers and the laws of indices

Key ideas

Watch out

Exercise 2G Laws of indices

Zero and negative indices

Key ideas

Watch out

Exercise 2H Zero and negative indices

Estimating roots and powers, and using a calculator

Key ideas

Watch out

Exercise 2I Estimating roots and powers (non-calculator)

Exercise 2J Powers and roots with a calculator

Chapter summary

Can you…?

Key terms

Exercise 2Z Check-up (non-calculator)

Exercise 2X Exam-style questions