YEAR 9 · INDICES AND SCIENTIFIC NOTATION · 15 MIN

Integer index laws

Explain and combine whole-number powers by tracking repeated factors, then check which law applies.

In this lesson

Identify the base and exponent.
Apply product, quotient and power-of-a-power laws.
Choose a law from the structure of an expression.

Before you start

Multiplication and division
Meaning of a power

01 / LEARN

Understand the idea.

Repeated factors explain the laws

An exponent counts repeated factors: $3^{4}$ contains four factors of $3$ , not $3$ multiplied by $4$ . Multiplying same-base powers joins the factor lists, so $a^{m} a^{n} = a^{m + n}$ . The base itself stays unchanged.

Dividing powers cancels matching factors, giving $a^{m} \div a^{n} = a^{m - n}$ for $a \neq 0$ . Check that the bases match before combining exponents; a power of $2$ and a power of $3$ cannot use this same-base rule.

\begin{array}{c} a^{m} a^{n} = a^{m + n} \\ \frac{a^{m}}{a^{n}} = a^{m - n} (a \neq 0) \end{array}

A power of a power

In a power of a power, the outside exponent repeats the entire inside power. There are $m$ factors in each of $n$ groups, so ${(a^{m})}^{n} = a^{m n}$ . Multiply the exponents because you are counting groups of factors.

Choose a law by the operation, not just the matching bases. Adding powers does not join factor lists: $2^{3} + 2^{4} = 8 + 16$ , rather than $2^{7}$ . Brackets also show whether a minus sign belongs to the base.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Combine multiplication and division

Write $2^{5} \times 2^{3} \div 2^{4}$ as one power and evaluate.

Worked solution

1
Combine the multiplication.
$2^{5} \times 2^{3} = 2^{5 + 3} = 2^{8}$
Same-base factors contribute a total of eight factors of 2.
2
Apply the quotient law.
$2^{8} \div 2^{4} = 2^{8 - 4} = 2^{4}$
Division by the nonzero power cancels four factors of 2.
3
Evaluate the remaining power.
$2^{4} = 2 \times 2 \times 2 \times 2 = 16$
The single-power form is 2 to the fourth power.

Answer

$16$

EXAMPLE 02

Use the outer exponent

Simplify ${(3^{2})}^{3} \div 3^{4}$ .

Worked solution

1
Simplify the power of a power.
${(3^{2})}^{3} = 3^{2 \times 3} = 3^{6}$
Three groups of two factors make six factors of 3.
2
Divide same-base powers.
$3^{6} \div 3^{4} = 3^{6 - 4} = 3^{2}$
The base is nonzero and four matching factors cancel.
3
Evaluate.
$3^{2} = 9$
Two factors of 3 multiply to 9.

Answer

$9$

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify ${(5^{2})}^{3} \div 5^{4}$ .

Show answer & reasoning

$25$

First ${(5^{2})}^{3} = 5^{2 \times 3} = 5^{6}$ . Then $5^{6} \div 5^{4} = 5^{6 - 4} = 5^{2} = 25$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Evaluate $4^{3}$ .
1 mark
Write $7^{2} \times 7^{4}$ as one power.
2 marks
Simplify $10^{8} \div 10^{5}$ and evaluate.
2 marks
Write ${(6^{3})}^{2}$ as one power.
2 marks
Simplify $2^{4} \times 2^{3} \div 2^{5}$ .
2 marks
Mia says $3^{2} + 3^{3} = 3^{5}$ . Explain the mistake and give the correct sum.
3 marks

Read the worked solutions

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

Evaluate $4^{3}$ .
1 mark
$4^{3} = 4 \times 4 \times 4 = 64$ .
$64$
Write $7^{2} \times 7^{4}$ as one power.
2 marks
The same-base product has exponent $2 + 4 = 6$ .
$7^{6}$
Simplify $10^{8} \div 10^{5}$ and evaluate.
2 marks
Subtract the exponents: $8 - 5 = 3$ .
$10^{3} = 1000$ .
$10^{3} = 1000$
Write ${(6^{3})}^{2}$ as one power.
2 marks
Multiply the exponents: $3 \times 2 = 6$ .
$6^{6}$
Simplify $2^{4} \times 2^{3} \div 2^{5}$ .
2 marks
Combine exponents: $4 + 3 - 5 = 2$ .
$2^{2} = 4$ .
$4$
Mia says $3^{2} + 3^{3} = 3^{5}$ . Explain the mistake and give the correct sum.
3 marks
The expression adds powers rather than multiplying them.
$3^{2} + 3^{3} = 9 + 27 = 36$ , not $3^{5} = 243$ .
$36$

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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.

Repeated factors explain the laws

A power of a power

See the method in action.

Combine multiplication and division

Combine the multiplication.

Apply the quotient law.

Evaluate the remaining power.

Use the outer exponent

Simplify the power of a power.

Divide same-base powers.

Evaluate.

Try it yourself.

Check your understanding

Keep practising