YEAR 9 · INDICES AND SCIENTIFIC NOTATION · 15 MIN

Zero/negative powers and reciprocals

Use zero and negative exponents as a consistent extension of the quotient law.

In this lesson

Explain why a non-zero number to power zero is 1.
Rewrite a negative power as a reciprocal.
Distinguish a negative exponent from a negative number.

Before you start

Same-base quotient law
Reciprocals and fractions

01 / LEARN

Understand the idea.

Why the zero power is one

For a nonzero base, dividing a power by itself gives $1$ . The quotient law also gives $a^{3} \div a^{3} = a^{0}$ , so $a^{0} = 1$ . The condition $a \neq 0$ matters because division by zero is undefined.

Lowering the exponent by one divides the value by the base. With base $2$ , the sequence $2^{2}, 2^{1}, 2^{0}, 2^{- 1}$ therefore has values $4, 2, 1, \frac{1}{2}$ . Negative powers continue the same pattern; they do not introduce a new sign rule.

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 $n$ , $a^{- n} = \frac{1}{a^{n}}$ . A negative exponent means reciprocal, not a negative result. With a positive base the reciprocal stays positive; with a negative base its sign depends on whether the power is odd or even.

Keep the base grouped when it is negative. The square in ${(- 2)}^{- 2} = \frac{1}{4}$ applies to the whole negative base. In $- 2^{- 2} = - \frac{1}{4}$ , the initial minus sign sits outside the power.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Evaluate a negative power

Evaluate $5^{- 3}$ .

Worked solution

1
Rewrite as a reciprocal.
$5^{- 3} = \frac{1}{5^{3}}$
The negative exponent places the corresponding positive power in the denominator.
2
Evaluate the denominator.
$5^{3} = 5 \times 5 \times 5 = 125$
The exponent counts three factors of 5.
3
State the exact fraction.
$5^{- 3} = \frac{1}{125}$
The reciprocal is positive because its base is positive.

Answer

$\frac{1}{125}$

EXAMPLE 02

A negative base with brackets

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

Worked solution

1
Keep the negative base inside brackets.
${(- 2)}^{- 3} = \frac{1}{{(- 2)}^{3}}$
The reciprocal rule preserves the complete base, including its sign.
2
Evaluate the odd power.
${(- 2)}^{3} = (- 2) (- 2) (- 2) = - 8$
An odd number of negative factors gives a negative product.
3
Simplify the reciprocal.
$\frac{1}{- 8} = - \frac{1}{8}$
A negative denominator can be written as a minus sign before the fraction.

Answer

$- \frac{1}{8}$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$0.0001$

$10^{- 4} = \frac{1}{10^{4}} = \frac{1}{10000} = 0.0001$ . The negative exponent makes a positive number smaller than 1.

Keep practising

6 worksheet questions · 11 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Why the zero power is one

Negative means reciprocal, not negative answer

See the method in action.

Evaluate a negative power

Rewrite as a reciprocal.

Evaluate the denominator.

State the exact fraction.

A negative base with brackets

Keep the negative base inside brackets.

Evaluate the odd power.

Simplify the reciprocal.

Try it yourself.

Check your understanding

Keep practising