Index laws

Write powers out as factors and the laws appear: multiplying powers of the same base adds the indices, dividing subtracts them, and a power of a power multiplies them. Anything (except 0) to the power 0 is 1.

2¹⁰ × 2¹⁰ has over a million as its answer, but you can write it down in a second: 2²⁰. Write the powers out once, and you will see why.

A power is repeated multiplication: $2^3 = 2 \times 2 \times 2$ . The base is 2 and the index (exponent) is 3. Writing powers out as factors gives the index laws for powers of the same base:

Law	Why	Example
$a^m \times a^n = a^{m+n}$	the factors join	$2^3 \times 2^2 = 2^5$
$a^m \div a^n = a^{m-n}$	pairs of factors cancel	$3^5 \div 3^2 = 3^3$
$(a^m)^n = a^{mn}$	$n$ groups of $m$ factors	$(2^3)^2 = 2^6$
$a^0 = 1$	$a^n \div a^n = 1$ and $= a^0$	$7^0 = 1$

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

2³ × 2⁴

Write each power out as 2s.

2³ × 2⁴ = 2 to what power?7112It is 4⁷

TryTry it on the board and compare the result with your prediction.

Join the factors

Try it yourselfYour edits are saved in this browser.

Make 2³ × 2⁴. How many 2s are there? Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Three laws and zero

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$3^5 \div 3^2 = \frac{3 \times 3 \times 3 \times 3 \times 3}{3 \times 3}$ Write both powers out.
$= 3 \times 3 \times 3 = 3^3$ Two pairs cancel: 5 − 2 = 3 factors left.
$(2^3)^2 = 2^3 \times 2^3 = 2^6$ A power of a power: 2 groups of 3 factors, 3 × 2.
$2^3 \div 2^3 = 1 = 2^0$ Everything cancels: so 2⁰ = 1.

Read Kai's working. What is the base?

5⁶ ÷ 5² = 5 to what power?

Kai simplifies 3² × 3⁴. Click the line where his working goes wrong.

Why is 5⁰ equal to 1?5³ ÷ 5³ = 1, and the quotient law gives 5³⁻³ = 5⁰Because any number to a power is at least 1It is a rule with no reasonBecause 0 × 5 = 0, and then add 1

ModifyChange the problem on purpose and see what happens.

Each law on the board

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Make 2⁶ ÷ 2². What is left?

Use the board. This ticks when you get there.

Make (2²)³. How many 2s?

Use the board. This ticks when you get there.

Show 5³ ÷ 5³. What is 5⁰?

Use the board. This ticks when you get there.

Which is bigger: (2³)² or 2³ × 2²?(2³)², which is 2⁶ = 642³ × 2², which is 2⁵2³ × 2², which is 2⁶They are equal

PractiseA few on your own, to check you have it.

Index laws

7⁵ × 7³ = 7 to what power?

(4²)⁵ = 4 to what power?

What is 10⁰?

Review: what is 2⁵?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

2⁹ two ways

Then write 2⁹ as a quotient of two powers of 2, and as a power of a power.

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Need a hint?

The indices must add to 9.

Another hint

With m bigger than n: 5 and 4, or 6 and 3.

Make a product of two powers of 2 equal to 2⁹, with the first index bigger than the second.

Use the board. This ticks when you get there.

For teachers

Students write powers out as factors on the board and discover the product, quotient and power laws and why a⁰ = 1 (AC9M8N02, node 1). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can explain each index law by writing the powers out.
I can use the laws to simplify powers of the same base.
I can explain why a⁰ = 1.

Before class

Do the Run board yourself first. Students need exponent notation (Number Foundations: Powers).

The one idea: the laws come from counting factors; the base never changes. The common mistake is multiplying the bases (Kai's mistake).

The board writes each power as a row of its factors: joined for a product, cancelled in pairs for a quotient, in groups for a power of a power. The quotient board keeps m at least n (negative indices are Year 9).

Wider: the pattern 2³, 2², 2¹, 2⁰ (halving each time) also gives 2⁰ = 1, and leads to 2⁻¹ = ½.

Running the lesson

Predict: ask a '12' student to write 2³ × 2⁴ out.
Run: ask what happens to the count when n goes up by 1.
Investigate: ask Kai how many 3s there are.
Modify: ask how (2²)³ differs from 2² × 2³.
Make: ask for all the ways they found.

The hard part (on purpose)

Keeping the base. Ask students to write the powers out as factors before they use any law.

When students get stuck

A student multiplies the bases. Ask: Write both powers out: what number is repeated? Fix: The base stays the same; count the factors.
A student multiplies indices for a product. Ask: How many factors altogether? Fix: Products add; powers of powers multiply.
A student says a⁰ = 0. Ask: What is 5³ ÷ 5³? Fix: 1, and the law calls it 5⁰.

Stretch and support

Stretch: Continue the pattern 2⁴, 2³, 2², 2¹, 2⁰ by halving. What should 2⁻¹ and 2⁻² be?

Support: A card for each law with the expanded-form picture beside it.

Where it leads

This starts AC9M8N02.

Where it leads: combining the laws (next lesson), scientific notation, and negative indices (Year 9).

What each step is for

2³ × 2⁴: Committing to the index of 2³ × 2⁴ shows whether they add, multiply, or multiply the bases too.
join the factors: Building products of powers and counting the factors shows the indices add while the base stays.
three laws and zero: Dividing cancels pairs, so indices subtract; the base never changes (Kai multiplies it); a⁰ = 1 follows from dividing a power by itself.
each law on the board: Building a quotient, a power of a power and a zero index on the board, then telling the product and power laws apart.
index laws: Each law once, with a zero index, mixed; multiplying the bases is the planted slip; the powers review keeps evaluating fresh.
2⁹ two ways: Finding a product that makes a target power uses the product law backwards.

Now you can

Explain each index law by writing the powers out.
Use the laws to simplify powers of the same base.
Explain why a⁰ = 1.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson6. Using the index lawsSimplify one step at a time: powers of powers first, then multiply and divide from left to right, keeping each base. The laws only join powers of the same base: 2³ × 3² cannot become one power. They also make scientific notation quick.

Source: content/maths/real-numbers/index-laws/lesson.md