year 8 maths

44 Chapter 1 Algebraic techniques 2 and indices
1J Index laws for multiplication and division
Recall that x 2 means x × x and x 3 means x × x × x . Index notation provides a convenient way to describe
repeated multiplication.
index or exponent or power
3 5 = 3 × 3 × 3 × 3 × 3
base
Notice that 3 5 × 3 2 = 3 × 3 × 3 × 3 × 3 × 3 × 3
3 5 3 2
which means that 3 5 × 3 2 = 3 7 .
Similarly it can be shown that 2 6 × 2 5 = 2 11 .
When dividing, note that:
5 10
5 = 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5
7 5 × 5 × 5 × 5 × 5 × 5 × 5
= 5 × 5 × 5
So 5 10 ÷ 5 7 = 5 3 .
L et’s start: Comparing powers
• Arrange these numbers from smallest to largest.
2 3 , 3 2 , 2 5 , 4 3 , 3 4 , 2 4 , 4 2 , 5 2 , 1 20
• Did you notice any patterns?
• If all the bases were negative, how would that change your arrangement from smallest to largest?
For example, 2 3 becomes (−2) 3 .
Key ideas
■
index or exponent or power
5 n = 5 × 5 × … × 5
base
The number 5 appears n times.
For example, 2 6 = 2 × 2 × 2 × 2 × 2 × 2 = 64.
■ An expression such as 4 × 5 3 can be written in expanded form as 4 × 5 × 5 × 5 .
■ The index law for multiplying terms with the same base: 3 m × 3 n = 3 m+n
For example, 3 4 × 3 2 = 3 6 .
■ The index law for dividing terms with the same base: 3 m ÷ 3 n = 3m
3 n = 3m−n
For example, 3 8 ÷ 3 5 = 3 3 .
Cambridge Maths NSW
Stage 4 Year 8 Second edition
ISBN 978-1-108-46627-1 © Palmer et al. 2018
Cambridge University Press
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