Every law is just counting copies

An index is shorthand for how many times a number has been multiplied by itself. Every law comes from writing the copies out and counting.

Law	Why it is true	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	Put two collections side by side and count	2³ × 2² = 2⁵ = 32
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Cancel matching copies and count what is left	2⁵ ÷ 2² = 2³ = 8
(aᵐ)ⁿ = aᵐⁿ	n groups of m copies each, so multiply	(2³)² = 2⁶ = 64
a⁰ = 1	A number divided by itself	2⁰ = 1

Why anything to the power zero is 1

16, 8, 4, 2, and each step halves, so the next must be 1. The division law agrees: 2³ ÷ 2³ is 2⁰, and a number over itself is 1.

2³ is not 2 times 3

2³ means 2 × 2 × 2 = 8, not 6. The index counts copies; it is not a multiplier.

Negative indices

A negative index does not make the answer negative; it puts the base underneath. 2⁻² is one quarter.

The laws only work with the same base

2³ × 3² cannot become a single power, because the bases differ. Work each out: 8 × 9 = 72.

Standard form

45000 becomes 4.5 × 10⁴ and 0.00032 becomes 3.2 × 10⁻⁴. The index records how many places the point moved.

Which way the sign goes

Big number, positive index. Small number, negative index. Check whether the answer should be larger or smaller than the front number.

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Worked Example 1

Simplify 3⁴ × 3³ ÷ 3⁵.

The bases are all the sameAll base 3, so the laws apply

Multiplying adds the indices3⁴ × 3³ = 3⁷

Dividing subtracts3⁷ ÷ 3⁵ = 3² = 9

Check the long way81 × 27 = 2187, and 2187 ÷ 243 = 9. Same answer, far less work.

Worked Example 2

Work out 2⁻³ and explain what the negative index does.

Follow the halving pattern8, 4, 2, 1, ½, ¼, ⅛

Read off the answer2⁻³ = 1/8

What the sign meansNot negative. The minus puts the base underneath: 1 over 2³.

A useful checkA negative index always gives an answer between 0 and 1 when the base is above 1.

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Activity 23.1

The folding paper

What you needA large sheet of thin paper, a ruler and a table to record each fold.

MethodFold the sheet in half, count the layers, and keep going for as long as the paper allows.

Then predictWrite the layer count as a power of 2, then predict twenty folds without folding anything.

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An index counts how many copies of the base are multiplied together.

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1What is an index, and why is 2³ equal to 8 rather than 6?

2Why do the indices add when you multiply and subtract when you divide?

3Why is anything to the power zero equal to 1 and not 0?

4What does a negative index mean, and why is the answer not negative?

5What is standard form for, and how do you decide the sign of the index?

Question 1 of 10

1What is 2³?

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Unit 23 · Indices

Indices

Mathematics · Grade 7

What You Will Be Able To Do

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Use the index laws, and rebuild any of them if you forget

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Explain why anything to the power zero is 1, in two ways

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Handle negative indices, and see why answers stay positive

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Write very large and very small numbers in standard form

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Every law in this unit comes from writing the copies out and counting them, so nothing here has to be taken on trust.

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Presenter notes: Open with a claim nobody believes. Say that a sheet of newspaper folded thirty times would be taller than the highest mountains, and take a vote on whether that could possibly be true. It is, and the class will find out why later in the lesson. Then write 2³ on the board and ask for its value, expecting some to say six.