YEAR 8 · INDEX LAWS · 16 MIN

Dividing same-base powers

Simplify a quotient of same-base powers by cancelling matching factors.

In this lesson

Apply the quotient law in the correct order.
Explain cancellation of equal factors.
Respect the nonzero-base condition.

Before you start

Exponent notation
Division and fractions

01 / LEARN

Understand the idea.

Cancel matching factors

Write a quotient of powers as a fraction and cancel equal factors. In $\frac{5^{6}}{5^{2}}$ , two factors of five cancel from each part, leaving four in the numerator. This explains why the denominator exponent is subtracted from the numerator exponent.

The common base must be nonzero, because a zero denominator is undefined. Keep the base unchanged and preserve the order of subtraction. These examples use a numerator exponent at least as large as the denominator exponent.

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

Check division is allowed

When the two nonzero powers are identical, all their factors cancel and the quotient is one. Nothing remaining in the numerator after cancellation means a factor of one remains; it does not mean the whole fraction becomes zero.

If unsure about a quotient law, evaluate manageable powers separately and divide their values. This gives a different route to check your simplified expression. A numerical check supports that example; cancellation explains why the general rule works.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A shorter power

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

Worked solution

1
Apply cancellation to the matching base.
$\frac{5^{6}}{5^{2}} = 5^{6 - 2}$
Two of the six numerator factors cancel with the two denominator factors.
2
Write the remaining power.
$5^{6 - 2} = 5^{4}$
Four factors of five remain after cancellation.
3
Evaluate the power.
$5^{4} = 625$
Multiplying four fives gives six hundred and twenty-five.

Answer

$5^{4} = 625$

EXAMPLE 02

All factors cancel

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

Worked solution

1
Evaluate the equal powers.
$2^{4} = 16$
Both the dividend and divisor have the same nonzero value.
2
Write the quotient.
$2^{4} \div 2^{4} = 16 \div 16$
Substituting the power values keeps the original division unchanged.
3
Divide equal nonzero values.
$16 \div 16 = 1$
One group of sixteen fits into sixteen.

Answer

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$8^{4}$

The base eight is nonzero and the bases match. Subtract the divisor's exponent: $8^{7} \div 8^{3} = 8^{7 - 3} = 8^{4}$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Cancel matching factors

Check division is allowed

See the method in action.

A shorter power

Apply cancellation to the matching base.

Write the remaining power.

Evaluate the power.

All factors cancel

Evaluate the equal powers.

Write the quotient.

Divide equal nonzero values.

Try it yourself.

Check your understanding

Keep practising