The Chemistry Maths Book, Second Edition

16 Chapter 1Numbers, variables, and units

The index rule

Numbers are often written in the form a

m

, where ais called the baseand mis the

indexor exponent; for example, 1001 = 110

2

with base 10 and exponent 2 , and 161 = 12

4

with base 2 and exponent 4. When mis a positive integer, a

m

is the mth power of a;

form 1 = 13 ,

a

3

1 = 1 a 1 × 1 a 1 × 1 a,(−a)

3

1 = 1 (−a) 1 × 1 (−a) 1 × 1 (−a) 1 = 1 (−1)

3

1 × 1 a

3

1 = 1 −a

3

Numbers are also defined with negative and non-integral exponent. In practice, the

numbera

m

is read ‘a to the power m’ or ‘a to the m’, even when mis not a positive

integer. The rule for the product of numbers in base–index form is

9.a

m

a

n

1 = 1 a

m+n

(index rule)

For example,

a

3

a

2

1 = 1 (a 1 × 1 a 1 × 1 a) 1 × 1 (a 1 × 1 a) 1 = 1 a 1 × 1 a 1 × 1 a 1 × 1 a 1 × 1 a 1 = 1 a

5

1 = 1 a

3 + 2

Three auxiliary rules are

10.a

m

2 a

n

1 = 1 a

m−n

11.(a

m

)

n

1 = 1 (a

n

)

m

1 = 1 a

m×n

12.(ab)

m

1 = 1 a

m

b

m

Rule 10 defines numbers with zero and negative exponents. Thus, settingm 1 = 1 n,

a

n

2 a

n

1 = 1 a

n−n

1 = 1 a

0

1 = 11

and any number raised to power zero is unity; for example, 2

3

22

3

1 = 12

3 − 3

1 = 12

0

1 = 11

because 2

3

22

3

1 = 11. Also, settingm 1 = 10 in rule 10 ,

a

0

2 a

n

1 = 112 a

n

1 = 1 a

−n

so that the inverse ofa

n

isa

−n

. In particular, 12 a 1 = 1 a

− 1

.

EXAMPLES 1.13The index rule

rule examples

9.a

m

a

n

1 = 1 a

m+n

(a) 2

3

1 × 12

2

1 = 12

3 + 2

1 = 12

5

(b) 3

6

1 × 13

− 3

1 = 13

6 − 3

1 = 13

3

(c) 2

122

1 × 12

124

1 = 12

122 + 124

1 = 12

324

10.a

m

2 a

n

1 = 1 a

m−n

(d) 2

324

22

124

1 = 12

324 − 124

1 = 12

122

(e) 2

4

22

− 2

1 = 12

4 −(−2)

1 = 12

4 + 2

1 = 12

6

(f ) 3

4

23

4

1 = 13

4 − 4

1 = 13

0

1 = 11