Zero/negative powers and reciprocals | Year 9

01 / LEARN

The idea, step by step

Why the zero power is one

For a non-zero base, $a^{3} \div a^{3} = 1$ . The index law also writes the quotient as $a^{3 - 3} = a^{0}$ , so $a^{0} = 1$ . This does not justify assigning a value to $0^{0}$ in this course.

Continue down a power table by dividing by the base each time. For base 2 the values $8, 4, 2, 1, \frac{1}{2}, \frac{1}{4}$ correspond to exponents $3, 2, 1, 0, - 1, - 2$ . A negative exponent keeps this pattern consistent.

A power table continues through zero
Exponent	3	2	1	0	−1	−2
Power of 2	8	4	2	1	1/2	1/4

Negative means reciprocal, not negative answer

For $a \neq 0$ and a positive integer $n$ , $a^{- n} = \frac{1}{a^{n}}$ . The exponent changes where the power sits in a fraction; it does not place a minus sign in front of its value.

Brackets matter for negative bases. ${(- 2)}^{- 2} = \frac{1}{{(- 2)}^{2}} = \frac{1}{4}$ , but $- 2^{- 2}$ means $- (2^{- 2}) = - \frac{1}{4}$ . Always preserve the base when rewriting a reciprocal.

A common trap

Reading $a^{- n}$ as $- a^{n}$ .

Instead: A negative exponent gives a reciprocal; the sign of the base still depends on the base and its power.

02 / WORKED EXAMPLES

See the thinking

Example 1

Evaluate a negative power

Evaluate $5^{- 3}$ .

$5^{- 3} = \frac{1}{5^{3}}$ .
$5^{3} = 125$ .
The reciprocal is $\frac{1}{125} = 0.008$ .

\frac{1}{125}

Example 2

A negative base with brackets

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

Rewrite the negative power: ${(- 2)}^{- 3} = \frac{1}{{(- 2)}^{3}}$ .
${(- 2)}^{3} = - 8$ because the exponent is odd.
$\frac{1}{- 8} = - \frac{1}{8}$ .

- \frac{1}{8}

Check your understanding

Evaluate $10^{- 4}$ as a decimal.

Show answer

$0.0001$

$10^{- 4} = \frac{1}{10^{4}} = \frac{1}{10000}$ .

03 / WORKSHEET

Your turn

6 questions · 11 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Evaluate $9^{0}$ .
1 mark
Write $4^{- 2}$ as a fraction.
2 marks
Evaluate ${(- 3)}^{- 2}$ .
2 marks
Write $0.000001$ as a power of 10.
2 marks
Evaluate $2^{3} \times 2^{- 5}$ .
2 marks
Explain why $0^{- 2}$ is not defined.
2 marks

Read the worked solutions

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

Evaluate $9^{0}$ .
1 mark
Any non-zero base raised to zero is 1.
$1$
Write $4^{- 2}$ as a fraction.
2 marks
$4^{- 2} = \frac{1}{4^{2}} = \frac{1}{16}$ .
$\frac{1}{16}$
Evaluate ${(- 3)}^{- 2}$ .
2 marks
${(- 3)}^{- 2} = \frac{1}{{(- 3)}^{2}} = \frac{1}{9}$ .
$\frac{1}{9}$
Write $0.000001$ as a power of 10.
2 marks
$0.000001 = \frac{1}{1000000} = \frac{1}{10^{6}} = 10^{- 6}$ .
$10^{- 6}$
Evaluate $2^{3} \times 2^{- 5}$ .
2 marks
Add exponents: $3 + (- 5) = - 2$ .
$2^{- 2} = \frac{1}{4}$ .
$\frac{1}{4}$
Explain why $0^{- 2}$ is not defined.
2 marks
$0^{- 2}$ would mean $\frac{1}{0^{2}} = \frac{1}{0}$ .
Division by zero is not defined.
It would require division by zero.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 numerical integer exponent laws and their extension to variables; scientific notation and practical measurement applications. Excludes fractional-index and full surd algebra taught in Year 10.

Official curriculum source ↗

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