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1.3 Indices

1.3 Indices

"is table shows powers of 3. Look at the patterns in the table. 3 is 3 to the power 4.
4
4 is called the index.
Power 3 −4 3 −3 3 −2 3 −1 3 0 3 1 3 2 3 3 3 4 3 5
The plural of index is indices.
1 1 1 1
Value 81 27 9 3 1 3 9 27 81 243

Negative powers of any positive integer are fractions. Here are some more examples.

2 = 2 × 2 × 2 × 2 = 16 2 = 1 7 = 7 × 7 × 7 = 353 7 = 1
4
−3
3
−4
16 343
Any positive integer to the power 0 is 1. 2 = 1 7 = 1 12 = 1
0
0
0
Worked example 1.3
Write these as fractions. a 2 b 6 −2
−6
a 2 = 1 = 1 2 = 2 × 2 × 2 × 2 × 2 × 2 = 64
6
−6
2 6 64
b 6 = 1 2 = 1 6 = 36
−2
2
6 36
) Exercise 1.3

−2
−1
1 Write each number as a fraction. a 5 b 5 c 5 d 5 −4
−3
2 Write each number as a fraction or as an integer.
a 7 b 7 c 7 d 7 e 7 3
−2
0
−1
2
3 Write each number as a fraction.
a 4 b 10 c 2 d 12 e 15 f 20 −2
−3
−2
−1
−1
−2
0
0
4 a Simplify each number. i 2 ii 5 iii 10 iv 20 0
0
b Write the results in part a as a generalised rule.
5 Write each expression as a single number.
a 2 + 2 + 2 b 3 + 3 + 3 + 3 c 5 − 5 − 5 −1
0
0
2
−1
−2
−1
0
6 Write each number as a decimal.
a 5 b 5 c 10 d 10 e 10 −3
−2
−1
−2
−1
7 Write each number as a power of 2.
a 8 b 1 c 1 d 1 e 1
2 4 16
10
8 2 = 1024. In computing this is called 1K. Write each of these as a power of 2.
a 2K b 0.5K c 1
1K
1 Integers, powers and roots 11