YEAR 8 · INDEX LAWS · 16 MIN

Zero index

Explain the zero-index rule for a nonzero base and use it carefully.

In this lesson

Use a quotient to justify a zero power.
Apply the rule only to nonzero bases.
Distinguish ${(- a)}^{0}$ from $- a^{0}$ .

Before you start

Quotient law
Order of operations

01 / LEARN

Understand the idea.

Why the value is one

Compare two descriptions of the same nonzero quotient: $\frac{9^{3}}{9^{3}} = 1$ , while the quotient law gives $9^{3 - 3} = 9^{0}$ . Therefore $9^{0} = 1$ . This reasoning extends to any nonzero base.

A zero exponent is different from multiplication by zero. The expression $9 \times 0$ is zero, but $9^{0}$ is one. The index law must keep a consistent value when equal nonzero powers divide.

a^{0} = 1 (a \neq 0)

Read signs and conditions

Brackets determine the base. In ${(- 5)}^{0}$ , the base is negative five, so the value is one. In $- 5^{0}$ , only five is raised to zero and the outside minus remains, giving negative one.

Always retain the condition $a \neq 0$ in the zero-index rule. The quotient argument cannot assign a value to $0^{0}$ , because it would require division by zero. This lesson's rule therefore applies only to nonzero bases.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Explain with a fraction

Explain why $6^{0} = 1$ .

Worked solution

1
Evaluate a nonzero power divided by itself.
$\frac{6^{2}}{6^{2}} = \frac{36}{36} = 1$
Equal nonzero numerator and denominator form a quotient of one.
2
Apply the quotient law to the same expression.
$\frac{6^{2}}{6^{2}} = 6^{2 - 2} = 6^{0}$
Cancelling equal factors subtracts the two exponents.
3
Equate the two descriptions.
$6^{0} = 1$
Both calculations describe the same valid quotient, so their values agree.

Answer

EXAMPLE 02

Two zero powers

Evaluate ${(- 5)}^{0} + 2^{0}$ .

Worked solution

1
Evaluate the bracketed negative base.
${(- 5)}^{0} = 1$
Negative five is nonzero, so the zero-index rule applies to the whole base.
2
Evaluate the second zero power.
$2^{0} = 1$
Two is also a nonzero base.
3
Add the values.
${(- 5)}^{0} + 2^{0} = 1 + 1 = 2$
The powers must be evaluated before the addition.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

Evaluate $11^{0} \times 3$ .

Show answer & reasoning

The nonzero base eleven gives $11^{0} = 1$ . Therefore $11^{0} \times 3 = 1 \times 3 = 3$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $7^{0}$ .
1 mark
Evaluate ${(- 8)}^{0}$ .
1 mark
Evaluate $- 8^{0}$ .
2 marks
Use a quotient to explain $6^{0} = 1$ .
2 marks
Evaluate $3^{0} + 4^{2}$ .
2 marks
A student uses $a^{0} = 1$ to claim $0^{0} = 1$ . Is this a valid use of the rule? Explain.
3 marks

Read the worked solutions

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

Evaluate $7^{0}$ .
1 mark
The base 7 is nonzero.
1
Evaluate ${(- 8)}^{0}$ .
1 mark
The base is the nonzero bracketed number -8.
1
Evaluate $- 8^{0}$ .
2 marks
$8^{0} = 1$ , then apply the outside minus sign.
-1
Use a quotient to explain $6^{0} = 1$ .
2 marks
A nonzero number divided by itself is 1.
$6^{3} \div 6^{3} = 1 = 6^{3 - 3} = 6^{0}$ .
Evaluate $3^{0} + 4^{2}$ .
2 marks
$1 + 16 = 17$ .
17
A student uses $a^{0} = 1$ to claim $0^{0} = 1$ . Is this a valid use of the rule? Explain.
3 marks
The quotient proof would divide zero by zero when the base is zero.
No. The rule requires a nonzero base.

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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.

Why the value is one

Read signs and conditions

See the method in action.

Explain with a fraction

Evaluate a nonzero power divided by itself.

Apply the quotient law to the same expression.

Equate the two descriptions.

Two zero powers

Evaluate the bracketed negative base.

Evaluate the second zero power.

Add the values.

Try it yourself.

Check your understanding

Keep practising