Multiplying same-base powers | Year 8

01 / LEARN

The idea, step by step

Join the factors

Multiplying $3^{2}$ by $3^{4}$ joins two factors of 3 to four more. There are six factors altogether, so the product is $3^{6}$ . The base stays 3. This explains why the exponents add when same-base powers are multiplied. Leave the result as one power unless evaluation is requested.

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

Read a missing exponent

A single factor such as 9 is $9^{1}$ . Writing that hidden exponent helps when combining powers. Check that the bases really match before using the rule: $2^{3} \times 3^{2}$ cannot be simplified by adding bases or exponents, because the factors are different. Evaluate the separate powers instead.

A common trap

$2^{3} \times 2^{4} = 4^{7}$ .

Instead: The base remains 2: the seven factors are all 2, giving $2^{7}$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Combine and evaluate

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

$3^{2} \times 3^{4} = 3^{2 + 4} = 3^{6}$ .
$3^{6} = 729$ .

3^{6} = 729

Example 2

Use an invisible exponent

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

$9 = 9^{1}$ .
$9^{3} \times 9^{1} = 9^{3 + 1} = 9^{4}$ .

9^{4}

Check your understanding

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

Show answer

$6^{5}$

The matching bases allow $2 + 3 = 5$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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

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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 ↗

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