Integer index laws | Year 9

01 / LEARN

The idea, step by step

Repeated factors explain the laws

In $3^{4}$ , the base is 3 and the exponent tells us that four factors of 3 are multiplied. An exponent is not a multiplier: $3^{4} = 81$ , whereas $3 \times 4 = 12$ .

Multiplying powers with the same base joins two lists of repeated factors, so add their exponents. Dividing cancels equal factors, so subtract the denominator exponent. For a quotient, the base must be non-zero.

a^{m} a^{n} = a^{m + n}, \frac{a^{m}}{a^{n}} = a^{m - n} (a \neq 0)

A power of a power

A power outside brackets repeats the entire inside power. There are $m$ factors in each of $n$ groups, giving $m n$ factors altogether: ${(a^{m})}^{n} = a^{m n}$ .

Check the operation before choosing a rule. $2^{3} + 2^{4}$ is addition, so the product law cannot combine it into a single power by adding exponents. Evaluate or factor instead. Brackets also decide whether a negative sign is part of the base.

A common trap

Adding exponents in a power of a power.

Instead: Multiply exponents for a power of a power; add only when multiplying same-base powers.

02 / WORKED EXAMPLES

See the thinking

Example 1

Combine multiplication and division

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

All factors have base 2; combine the product to get $2^{5 + 3} = 2^{8}$ .
Dividing by $2^{4}$ gives $2^{8 - 4} = 2^{4}$ .
$2^{4} = 16$ .

16

Example 2

Use the outer exponent

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

The outer power repeats $3^{2}$ three times, so ${(3^{2})}^{3} = 3^{6}$ .
$3^{6} \div 3^{4} = 3^{6 - 4} = 3^{2}$ .
$3^{2} = 9$ .

9

Check your understanding

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

Show answer

$25$

The power of a power gives $5^{6}$ ; division leaves $5^{2} = 25$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 $3^{5} = 243$ .
$36$

Checking saved progress…

Continue learning

Zero/negative powers and reciprocals

Next lesson

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 ↗

Back to Indices and scientific notation