Prime factorisation and exponent notation | Year 7

01 / LEARN

The idea, step by step

Keep splitting composite factors

A prime factorisation expresses a whole number greater than 1 as a product of primes. Split the number into a factor pair and continue splitting any composite factors. Stop only when every factor is prime.

Different splitting choices give the same prime factors. For example, splitting 24 as $6 \times 4$ or $3 \times 8$ eventually gives three factors of 2 and one factor of 3.

A power counts equal factors

In $2^{4}$ , 2 is the base and 4 tells us that four copies of 2 are multiplied. It does not mean $2 \times 4$ . Repeated prime factors can be grouped into powers.

Prime factorizations also reveal common factors. To build a common factor, use only primes shared by both numbers, and no more copies than either number contains.

72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^{3} \times 3^{2}

A common trap

$3^{4}$ means $3 \times 4$ .

Instead: The exponent counts repeated multiplication: $3^{4} = 3 \times 3 \times 3 \times 3 = 81$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Prime factorisation of 72

Write 72 as a product of prime powers.

Split $72 = 8 \times 9$ .
Then $8 = 2 \times 2 \times 2$ and $9 = 3 \times 3$ .
Group repeated factors into powers.

2^{3} \times 3^{2}

Example 2

Use prime factors to find an HCF

Find the highest common factor of 18 and 24.

Write $18 = 2 \times 3^{2}$ and $24 = 2^{3} \times 3$ .
Both contain one factor of 2 and at least one factor of 3.
The greatest shared product is $2 \times 3 = 6$ .

Check your understanding

Write $5 \times 5 \times 5 \times 2$ using a power.

Show answer

$2 \times 5^{3}$

There are three factors of 5 and one factor of 2.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write 60 as a product of prime powers.
2 marks
Write 108 as a product of prime powers.
2 marks
Write a product of five 7s using exponent notation, then evaluate the power.
2 marks
Evaluate $2^{3} \times 5^{2}$ .
2 marks
Use prime factors to find the highest common factor of 48 and 72.
3 marks
Find the whole-number value of $n$ if $2^{n} \times 3 = 96$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write 60 as a product of prime powers.
2 marks
Split $60 = 6 \times 10$ .
Then $60 = (2 \times 3) (2 \times 5) = 2^{2} \times 3 \times 5$ .
$2^{2} \times 3 \times 5$
Write 108 as a product of prime powers.
2 marks
Split $108 = 4 \times 27$ .
Then $4 = 2^{2}$ and $27 = 3^{3}$ .
$2^{2} \times 3^{3}$
Write a product of five 7s using exponent notation, then evaluate the power.
2 marks
Five equal factors give $7^{5}$ .
Multiply $7 \times 7 \times 7 \times 7 \times 7 = 16.807$ .
$7^{5} = 16.807$
Evaluate $2^{3} \times 5^{2}$ .
2 marks
Calculate $2^{3} = 8$ and $5^{2} = 25$ .
Then $8 \times 25 = 200$ .
200
Use prime factors to find the highest common factor of 48 and 72.
3 marks
$48 = 2^{4} \times 3$ and $72 = 2^{3} \times 3^{2}$ .
Use three factors of 2 and one factor of 3.
$2^{3} \times 3 = 24$ .
24
Find the whole-number value of $n$ if $2^{n} \times 3 = 96$ .
3 marks
Divide both sides by 3 to get $2^{n} = 32$ .
$32 = 2^{5}$ , so $n = 5$ .
$n = 5$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Place value and powers of ten; efficient whole-number calculations and order of operations; factors, multiples and primes; prime factorisation and exponent notation; square numbers and square roots; estimation and rounding.

Official curriculum source ↗

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