YEAR 8 · INDEX LAWS · 16 MIN

Powers of powers

Treat a power raised to another power as repeated groups of equal factors.

In this lesson

Multiply exponents in a power of a power.
Distinguish this law from the product law.
Keep a bracketed negative base intact.

Before you start

Exponent notation
Multiplying same-base powers

01 / LEARN

Understand the idea.

Groups of groups

A power of a power repeats the entire inner power. In ${(2^{3})}^{4}$ , there are four groups, each containing three factors of two. Counting all the factors gives $3 \times 4 = 12$ , so the result is $2^{12}$ .

The multiplication of exponents follows from grouping factors, rather than from a new rule for the base. With positive whole-number exponents, ${(a^{m})}^{n}$ contains $n$ groups of $m$ factors, giving $a^{m n}$ .

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

Choose by structure

Read the structure before choosing an index law. In $2^{3} \times 2^{4}$ , add the exponents because two products are joined. In ${(2^{3})}^{4}$ , multiply them because the whole inner product is repeated four times.

Keep brackets around a negative base. An even number of negative factors gives a positive result; an odd number gives a negative result. The brackets in ${(- 3)}^{6}$ tell you that the negative sign is part of every factor.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Four groups of three

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

Worked solution

1
Expand the outer repetition.
${(2^{3})}^{4} = 2^{3} \times 2^{3} \times 2^{3} \times 2^{3}$
The outer exponent four means four copies of the complete inner power.
2
Count the factors of two.
${(2^{3})}^{4} = 2^{3 \times 4} = 2^{12}$
Four groups of three factors contain twelve factors altogether.
3
Evaluate the resulting power.
$2^{12} = 4096$
The expression is twelve copies of two multiplied together.

Answer

$2^{12} = 4096$

EXAMPLE 02

An even final exponent

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

Worked solution

1
Combine the nested powers.
$({(- 3)}^{2})^{3} = {(- 3)}^{2 \times 3}$
The outer power repeats a pair of negative-three factors three times.
2
Simplify the exponent.
${(- 3)}^{2 \times 3} = {(- 3)}^{6}$
Three pairs contain six factors of negative three.
3
Determine the sign and value.
${(- 3)}^{6} = 3^{6} = 729$
Six is even, so the negative factors pair to produce a positive result.

Answer

729

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$7^{6}$

Two groups each contain three factors of seven. Therefore ${(7^{3})}^{2} = 7^{3 \times 2} = 7^{6}$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Groups of groups

Choose by structure

See the method in action.

Four groups of three

Expand the outer repetition.

Count the factors of two.

Evaluate the resulting power.

An even final exponent

Combine the nested powers.

Simplify the exponent.

Determine the sign and value.

Try it yourself.

Check your understanding

Keep practising