Foundation and Higher · index laws

Index laws.
Add, subtract,
multiply.

same base,
then look at the powers

another word for power —

Index

in a³ the base is a, the index is 3

the laws need the same base

law 1 of 4

Multiplying: add

1. Three lots of a, then four more.

2. So add the indices: 3 + 4 = 7.

✓ a^m × aⁿ = a^{m + n}

a³ × a⁴

= (aaa) × (aaaa) seven of them

= a^{3 + 4} = a⁷

law 2 of 4

Dividing: subtract

1. Two of the six cancel out.

2. So subtract the indices: 6 − 2 = 4.

✓ a^m ÷ aⁿ = a^{m − n}

a⁶ ÷ a²

= a^{6 − 2}

= a⁴

law 3 of 4

Power of a power

1. Squared means two lots of a³.

2. So multiply the indices: 3 × 2 = 6.

✓ (a^m)ⁿ = a^mn

(a³)²

= a³ × a³

= a^{3 × 2} = a⁶

law 4 of 4

Zero and negative

1. Anything divided by itself is 1.

2. A negative index means “one over”.

✓ 2⁻³ = 18, not −8

a³ ÷ a³ = a⁰ = 1

a² ÷ a⁵ = a⁻³

a⁻³ = 1a³

index laws, same base —

Multiply

add the indices

Divide

subtract the indices

Brackets

multiply the indices

Numbers in front multiply as normal:
2a³ × 5a⁴ = 10a⁷.

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