Combining numerical index laws | Year 8

01 / LEARN

Understand the idea.

Write a plan

Before calculating, identify each operation on the powers. Resolve a power of a power by multiplying its exponents; combine matching-base products by adding exponents and quotients by subtracting. Keep the original grouping visible as you work.

A single exponent calculation can organise a fraction whose factors all share one nonzero base. Ordinary sums remain separate: $2^{3} + 2^{4}$ requires addition of values, because the product law does not apply to an addition sign.

Check without repeating the shortcut

Check a manageable result by evaluating the original powers and comparing the numerical quotient. This uses a different route and can expose a wrong exponent operation. Merely repeating the same shortcut may repeat the same mistake.

Keep the answer in the requested form: one power, a numerical value, or both. These examples use nonzero numerical bases and nonnegative final exponents. Check the divisor before simplifying, since an index law cannot make division by zero valid.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Power, product and quotient

Simplify $\frac{{(2^{3})}^{2} \times 2^{4}}{2^{5}}$ and evaluate.

Worked solution

1
Simplify the power of a power.
${(2^{3})}^{2} = 2^{3 \times 2} = 2^{6}$
Two groups of three factors contain six factors of two.
2
Join the numerator factors.
$2^{6} \times 2^{4} = 2^{6 + 4} = 2^{10}$
Multiplying powers with the same base adds their factor counts.
3
Cancel the denominator factors.
$\frac{2^{10}}{2^{5}} = 2^{10 - 5} = 2^{5}$
Five numerator factors cancel with the five denominator factors.
4
Evaluate the final power.
$2^{5} = 32$
The five remaining factors of two multiply to thirty-two.

Answer

$2^{5} = 32$

EXAMPLE 02

A different base

Simplify $\frac{3^{2} \times {(3^{3})}^{2}}{3^{5}}$ and evaluate.

Worked solution

1
Simplify the nested power.
${(3^{3})}^{2} = 3^{3 \times 2} = 3^{6}$
The outer square repeats the three-factor group twice.
2
Combine the product in the numerator.
$3^{2} \times 3^{6} = 3^{2 + 6} = 3^{8}$
All eight numerator factors have the same base three.
3
Divide the matching-base powers.
$\frac{3^{8}}{3^{5}} = 3^{8 - 5} = 3^{3}$
Cancelling five factors leaves three factors in the numerator.
4
Evaluate the remaining power.
$3^{3} = 27$
Three copies of three have product twenty-seven.

Answer

$3^{3} = 27$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Cancel two of the three factors of four: $4^{3} \div 4^{2} = 4^{3 - 2} = 4^{1} = 4$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Simplify $\frac{{(5^{2})}^{3}}{5^{4}}$ and evaluate.
2 marks
Simplify $\frac{2^{3} \times 2^{4}}{2^{5}}$ and evaluate.
2 marks
Simplify $\frac{{(10^{2})}^{3} \times 10^{0}}{10^{4}}$ and evaluate.
2 marks
Simplify $\frac{{(4^{3} \div 4)}^{2}}{4^{2}}$ and evaluate.
3 marks
Evaluate ${(6^{2})}^{2} \div 6^{4}$ .
2 marks
Explain the error in ${(2^{3})}^{2} = 2^{5}$ and give the correct value.
3 marks

Read the worked solutions

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

Simplify $\frac{{(5^{2})}^{3}}{5^{4}}$ and evaluate.
2 marks
The exponent is $2 \times 3 - 4 = 2$ .
$5^{2} = 25$
Simplify $\frac{2^{3} \times 2^{4}}{2^{5}}$ and evaluate.
2 marks
$3 + 4 - 5 = 2$ .
$2^{2} = 4$
Simplify $\frac{{(10^{2})}^{3} \times 10^{0}}{10^{4}}$ and evaluate.
2 marks
$2 \times 3 + 0 - 4 = 2$ .
$10^{2} = 100$
Simplify $\frac{{(4^{3} \div 4)}^{2}}{4^{2}}$ and evaluate.
3 marks
$4^{3} \div 4 = 4^{2}$ .
${(4^{2})}^{2} \div 4^{2} = 4^{4 - 2} = 4^{2} = 16$ .
$4^{2} = 16$
Evaluate ${(6^{2})}^{2} \div 6^{4}$ .
2 marks
$6^{4 - 4} = 6^{0} = 1$ .
1
Explain the error in ${(2^{3})}^{2} = 2^{5}$ and give the correct value.
3 marks
${(2^{3})}^{2} = 2^{3 \times 2} = 2^{6} = 64$ .
The exponents multiply; the correct value is 64.

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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.

Write a plan

Check without repeating the shortcut

See the method in action.

Power, product and quotient

Simplify the power of a power.

Join the numerator factors.

Cancel the denominator factors.

Evaluate the final power.

A different base

Simplify the nested power.

Combine the product in the numerator.

Divide the matching-base powers.

Evaluate the remaining power.

Try it yourself.

Check your understanding

Keep practising