YEAR 8 · INDEX LAWS · 16 MIN

Exponent notation review

Read a power as equal factors, then use brackets to identify its base.

In this lesson

Identify the base and exponent.
Evaluate small numerical powers.
Distinguish a negative base from a minus sign outside the power.

Before you start

Multiplication and division facts.
Reading exponent notation.

01 / LEARN

Understand the method.

THE CORE RULE

3^{4} = 3 \times 3 \times 3 \times 3

The exponent counts equal factors. It does not multiply the base.

Count the factors

The base is the repeated factor. The exponent says how many times that factor appears.

In $3^{4}$ , the base is $3$ and the exponent is $4$ . A square has exponent $2$ ; a cube has exponent $3$ . The first power of a number is the number itself.

5^{1} = 5, 5^{2} = 5 \times 5, 5^{3} = 5 \times 5 \times 5

Check what the exponent belongs to

In ${(- 2)}^{4}$ , the whole bracket is the base. Four negative factors multiply to a positive result.

In $- 2^{4}$ , calculate the power of $2$ first. The minus sign is outside that power, so the result is negative.

\begin{aligned} {(- 2)}^{4} & = 16 \\ - 2^{4} & = - 16 \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Evaluate a positive power

Evaluate $3^{4}$ .

Worked solution

1
Write four equal factors.
$3^{4} = 3 \times 3 \times 3 \times 3$
The exponent $4$ counts the factors of $3$ .
2
Multiply the factors.
$3^{4} = 9 \times 9 = 81$
Pairing the factors makes the multiplication easier.

Answer

$81$ .

EXAMPLE 02

Compare a negative base and an outside minus

Evaluate ${(- 2)}^{4}$ and $- 2^{4}$ .

Worked solution

1
Expand the bracketed power.
$\begin{aligned} {(- 2)}^{4} & = (- 2) (- 2) (- 2) (- 2) \\ = 16 \end{aligned}$
The minus sign is part of every repeated factor.
2
Apply the outside minus after evaluating the power.
$\begin{aligned} - 2^{4} & = - (2 \times 2 \times 2 \times 2) \\ = - 16 \end{aligned}$
Here the exponent applies to $2$ only.

Answer

${(- 2)}^{4} = 16$ , whereas $- 2^{4} = - 16$ .

03 / YOUR TURN

Try it yourself.

Calculate, explain, then apply.

CHECK 01

Write the power

Write $4 \times 4 \times 4 \times 4 \times 4$ as a power.

Show answer & reasoning

$4^{5}$ .

The base is $4$ , and it occurs as a factor five times.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $2^{5}$ .
2 marks
Evaluate $5^{3}$ .
2 marks
Evaluate ${(- 3)}^{2}$ .
2 marks
Evaluate $- 3^{2}$ .
2 marks
Write $7 \times 7 \times 7$ as a power.
1 mark
Which is larger: $2^{6}$ or $6^{2}$ ? Show both values.
3 marks

Read the worked solutions

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

Evaluate $2^{5}$ .
2 marks
$2 \times 2 \times 2 \times 2 \times 2 = 32$ .
32
Evaluate $5^{3}$ .
2 marks
$5 \times 5 \times 5 = 125$ .
125
Evaluate ${(- 3)}^{2}$ .
2 marks
$(- 3) (- 3) = 9$ .
9
Evaluate $- 3^{2}$ .
2 marks
$- (3 \times 3) = - 9$ .
-9
Write $7 \times 7 \times 7$ as a power.
1 mark
There are three equal factors of 7.
$7^{3}$
Which is larger: $2^{6}$ or $6^{2}$ ? Show both values.
3 marks
$2^{6} = 64$ and $6^{2} = 36$ .
$2^{6} = 64$ is larger than $6^{2} = 36$ .

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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 ↗