Page 16 - 新思维数学学生用书9 样章

1.3 Indices

1.3 Indices

In this section you will …

• use positive, negative and zero indices
• use index laws for multiplication and division.

This table shows powers of 3.
Tip

3 2 3 3 3 4 3 5 3 6
9 27 81 243 729 The index is
the small red
When you move one column to the right, the index increases by 1 and number.
the number multiplies by 3.
9 × 3 = 27 27 × 3 = 81 81 × 3 = 243, and so on.
When you move one column to the left, the index decreases by 1 and the

number divides by 3. You can use this fact to extend the table to the left:

3 −4 3 −3 3 −2 3 −1 3 0 3 1 3 2 3 3 3 4 3 5 3 6

1 1 1 1
81 27 9 3 1 3 9 27 81 243 729

1
1
1
9 ÷ 3 = 3 3 ÷ 3 = 1 1 3 = 1 ÷ 3 = ÷ = 1 , and so on.
3
÷
3 3 9 9 27
You can see from the table that 3 = 3 and 3 = 1. Tip
0
1
1
−
−
3
Also: 3 = 3 − 2 = 1 3 = 1 , and so on. 3 = 1 seems
1
0
3 3 2 3 3
In general, if n is a positive integer then 3 −n = 1 . These results are not strange but it fits
3 n the pattern.
only true for powers of 3. They apply to any positive integer.
−
0
3
For example: 5 − 2 = 1 = 1 8 = 1 = 1 6 = 1
5 2 25 8 3 512
In general, if a and n are positive integers then a = 1 and a − n = 1 .
0
a n
Exercise 1.3
1 Write each number as a fraction.
a 4 −1 b 2 −3 c 9 −2
d 6 −3 e 10 −4 f 2 −5
2 Here are five numbers: 2 3 4 5 6 0
−3
−4
−1
−2
List the numbers in order of size, smallest first.

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