YEAR 8 · INDEX LAWS · 16 MIN

Multiplying same-base powers

Combine products of same-base powers by counting their total equal factors.

In this lesson

Apply the same-base product rule.
Keep the base unchanged.
Recognise when the rule does not apply.

Before you start

Multiplication and division facts
Reading exponent notation

01 / LEARN

Understand the idea.

Join the factors

A positive whole-number exponent counts equal factors. Multiplying $3^{2}$ by $3^{4}$ joins two factors of three to four more. The result contains six factors of three, so the base stays the same and the exponents add.

Use the product law only when the bases match and the powers are multiplied. The expression $2^{3} + 2^{4}$ is a sum, so evaluate or add its values; counting joined factors does not describe that operation.

a^{m} \times a^{n} = a^{m + n}

Read a missing exponent

A single copy of a number has exponent one: $9 = 9^{1}$ . Make this hidden exponent visible before combining it with another power of nine. Exponents count factors, so one extra factor increases the exponent by one.

Check the bases before choosing a shortcut. In $2^{3} \times 3^{2}$ , the factors are different, so adding the exponents is not valid. If asked for one power, keep that form; evaluate it only when requested.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Combine and evaluate

Write $3^{2} \times 3^{4}$ as one power and evaluate.

Worked solution

1
Recognise the matching base.
$3^{2} \times 3^{4} = 3^{2 + 4}$
Multiplication joins two and four copies of the same factor three.
2
Combine the exponent.
$3^{2 + 4} = 3^{6}$
There are six factors altogether and the base remains three.
3
Evaluate the power.
$3^{6} = 729$
Six factors of three give seven hundred and twenty-nine.

Answer

$3^{6} = 729$

EXAMPLE 02

Use an invisible exponent

Write $9^{3} \times 9$ as one power.

Worked solution

1
Show the hidden exponent.
$9^{3} \times 9 = 9^{3} \times 9^{1}$
A single nine is one factor of nine.
2
Count the combined factors.
$9^{3} \times 9^{1} = 9^{3 + 1}$
The product contains three factors followed by one more.
3
State the single power.
$9^{3 + 1} = 9^{4}$
Four copies of nine are represented by exponent four.

Answer

$9^{4}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $6^{2} \times 6^{3}$ .

Show answer & reasoning

$6^{5}$

Both factors have base six, so join their factor counts: $6^{2} \times 6^{3} = 6^{2 + 3} = 6^{5}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Write $5^{2} \times 5^{3}$ as one power.
2 marks
Write $2^{4} \times 2^{2}$ as one power, then evaluate.
2 marks
Write $7 \times 7^{3}$ as one power.
2 marks
Write $10^{3} \times 10^{2}$ as one power.
2 marks
Write $4^{2} \times 4$ as one power.
2 marks
Find $m$ if $3^{2} \times 3^{m} = 3^{7}$ .
3 marks

Read the worked solutions

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

Write $5^{2} \times 5^{3}$ as one power.
2 marks
$2 + 3 = 5$ .
$5^{5}$
Write $2^{4} \times 2^{2}$ as one power, then evaluate.
2 marks
$2^{4 + 2} = 2^{6} = 64$ .
$2^{6} = 64$
Write $7 \times 7^{3}$ as one power.
2 marks
$7^{1} \times 7^{3} = 7^{1 + 3}$ .
$7^{4}$
Write $10^{3} \times 10^{2}$ as one power.
2 marks
$3 + 2 = 5$ .
$10^{5}$
Write $4^{2} \times 4$ as one power.
2 marks
$4^{2} \times 4^{1} = 4^{2 + 1}$ .
$4^{3}$
Find $m$ if $3^{2} \times 3^{m} = 3^{7}$ .
3 marks
$2 + m = 7$ , so $m = 5$ .
5

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Exponent notation review; multiplying same-base powers; dividing same-base powers; powers of powers; zero index; combining numerical index laws. Negative indices and general variable index laws become Year 9 core.

Official curriculum source ↗

In this lesson

Understand the idea.

Join the factors

Read a missing exponent

See the method in action.

Combine and evaluate

Recognise the matching base.

Combine the exponent.

Evaluate the power.

Use an invisible exponent

Show the hidden exponent.

Count the combined factors.

State the single power.

Try it yourself.

Check your understanding

Keep practising