Dividing same-base powers | Year 8

01 / LEARN

The idea, step by step

Cancel matching factors

In $5^{6} \div 5^{2}$ , cancel two factors of 5 from numerator and denominator. Four factors remain, giving $5^{4}$ . For a nonzero base, subtract the denominator exponent from the numerator exponent. The order matters: reversing the subtraction changes the result.

a^{m} \div a^{n} = a^{m - n} (a \neq 0)

Check division is allowed

Division by a zero base is not allowed. Here examples have nonnegative final powers; negative powers are developed later. Dividing any nonzero power by itself gives 1, because numerator and denominator are equal. Check a small example by evaluating both powers if you are unsure about the law.

A common trap

The exponents are divided when powers are divided.

Instead: Cancellation subtracts exponents: $2^{8} \div 2^{2} = 2^{6}$ , not $2^{4}$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A shorter power

Simplify $5^{6} \div 5^{2}$ and evaluate.

The exponent is $6 - 2 = 4$ .
$5^{6} \div 5^{2} = 5^{4} = 625$ .

5^{4} = 625

Example 2

All factors cancel

Evaluate $2^{4} \div 2^{4}$ .

$2^{4} = 16$ .
$16 \div 16 = 1$ .

Check your understanding

Simplify $8^{7} \div 8^{3}$ .

Show answer

$8^{4}$

$7 - 3 = 4$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $2^{8} \div 2^{3}$ and evaluate.
2 marks
Write $10^{6} \div 10^{2}$ as one power.
2 marks
Evaluate $6^{5} \div 6^{4}$ .
2 marks
Evaluate $7^{3} \div 7^{3}$ .
2 marks
Simplify $3^{7} \div (3^{2} \times 3^{3})$ and evaluate.
3 marks
Find $n$ if $8^{n} \div 8^{2} = 8^{4}$ .
3 marks

Read the worked solutions

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

Simplify $2^{8} \div 2^{3}$ and evaluate.
2 marks
$8 - 3 = 5$ .
$2^{5} = 32$
Write $10^{6} \div 10^{2}$ as one power.
2 marks
$6 - 2 = 4$ .
$10^{4}$
Evaluate $6^{5} \div 6^{4}$ .
2 marks
$6^{5 - 4} = 6^{1} = 6$ .
6
Evaluate $7^{3} \div 7^{3}$ .
2 marks
The same nonzero number is divided by itself.
1
Simplify $3^{7} \div (3^{2} \times 3^{3})$ and evaluate.
3 marks
The denominator is $3^{5}$ .
$3^{7 - 5} = 3^{2} = 9$ .
$3^{2} = 9$
Find $n$ if $8^{n} \div 8^{2} = 8^{4}$ .
3 marks
$n - 2 = 4$ , so $n = 6$ .
6

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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 ↗

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