Powers of powers | Year 8 | Ascent Library

01 / LEARN

The idea, step by step

Groups of groups

In ${(2^{3})}^{4}$ , the inner power $2^{3}$ is multiplied by itself four times. Each group has three factors of 2, so there are twelve factors altogether. This gives $2^{12}$ . Multiplying the exponents counts all the factors; adding them would count something different.

{(a^{m})}^{n} = a^{m n}

Choose by structure

Two powers side by side use the product law: $2^{3} \times 2^{4} = 2^{7}$ . A power around a power uses multiplication of exponents: ${(2^{3})}^{4} = 2^{12}$ . Look at the brackets, not just the numbers. If the bracketed base is negative, preserve the brackets until the sign of the final power is clear.

A common trap

${(3^{2})}^{4} = 3^{6}$ .

Instead: Four groups of two factors give exponent $2 \times 4 = 8$ , not 6.

02 / WORKED EXAMPLES

See the thinking

Example 1

Four groups of three

Write ${(2^{3})}^{4}$ as one power and evaluate.

${(2^{3})}^{4} = 2^{3 \times 4} = 2^{12}$ .
$2^{12} = 4096$ .

2^{12} = 4096

Example 2

An even final exponent

Evaluate ${({(- 3)}^{2})}^{3}$ .

${({(- 3)}^{2})}^{3} = {(- 3)}^{2 \times 3} = {(- 3)}^{6}$ .
Six negative factors give a positive product of 729.

729

Check your understanding

Write ${(7^{3})}^{2}$ as one power.

Show answer

$7^{6}$

$3 \times 2 = 6$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Write ${(5^{2})}^{3}$ as one power.
2 marks
Write ${(3^{4})}^{2}$ as one power.
2 marks
Write ${(10^{3})}^{2}$ as one power.
2 marks
Evaluate ${(2^{2})}^{3}$ .
2 marks
Evaluate ${({(- 2)}^{3})}^{2}$ .
2 marks
Find $n$ if ${(4^{2})}^{n} = 4^{8}$ .
3 marks

Read the worked solutions

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

Write ${(5^{2})}^{3}$ as one power.
2 marks
$2 \times 3 = 6$ .
$5^{6}$
Write ${(3^{4})}^{2}$ as one power.
2 marks
$4 \times 2 = 8$ .
$3^{8}$
Write ${(10^{3})}^{2}$ as one power.
2 marks
$3 \times 2 = 6$ .
$10^{6}$
Evaluate ${(2^{2})}^{3}$ .
2 marks
$2^{2 \times 3} = 2^{6} = 64$ .
64
Evaluate ${({(- 2)}^{3})}^{2}$ .
2 marks
${(- 2)}^{3 \times 2} = {(- 2)}^{6} = 64$ .
64
Find $n$ if ${(4^{2})}^{n} = 4^{8}$ .
3 marks
$2 n = 8$ , so $n = 4$ .
4

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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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