The Chemistry Maths Book, Second Edition

1.6 The algebra of real numbers 17

11.(a

m

)

n

1 = 1 (a

n

)

m

1 = 1 a

m×n

(g) (2

2

)

3

1 = 1 (2

2

) 1 × 1 (2

2

) 1 × 1 (2

2

) 1 = 12

2 × 3

1 = 12

6

(h) (2

122

)

2

1 = 12

(1 2 2)× 2

1 = 12

1

1 = 12

(i) (2

3

)

423

1 = 1 (2

423

)

3

1 = 12

3 × 423

1 = 12

4

(j)

12.(ab)

m

1 = 1 a

m

b

m

(k) (2 1 × 1 3)

2

1 = 12

2

1 × 13

2

(l) (−8)

123

1 = 1 (−1)

123

1 × 18

123

1 = 1 (−1) 1 × 121 = 1 − 2

0 Exercises 50–65

Example 1.13(h) shows that 2

122

1 × 12

122

1 = 12. It follows that , the square root

of 2. In general, for positive integer m, a

12 m

is the mth root of a:

Thus, 2

123

is a cube root of 2 because (2

123

)

3

1 = 12

(1 2 3)× 3

1 = 12

1

1 = 12. More generally, for

rational exponentm 2 n, rule 11 gives

a

m 2 n

1 = 1 (a

m

)

12 n

1 = 1 (a

12 n

)

m

or, equivalently,

so thata

m 2 n

is both the nth root of the mth power of a and the mth power of the nth

root. For example,

4

322

1 = 1 (4

3

)

122

1 = 1 (4

122

)

3

1 = 18

Although the index rules have been demonstrated only for integral and rational

indices, they apply to all numbers written in the base–index form. When the exponent

mis a variable,a

m

is called an exponential function (see Section 3.6 for real exponents

and Chapter 8 for complex exponents). Ifx 1 = 1 a

m

thenm 1 = 1 log

a

xis the logarithm of

x to base a(see Section 3.7).

Rules of precedence for arithmetic operations

An arithmetic expression such as

21 + 131 × 14

is ambiguous because its value depends on the order in which the arithmetic

operations are applied. The expression can be interpreted in two ways:

(2 1 + 1 3) 1 × 141 = 151 × 141 = 120

aaa

mn n m n m

==()

aa

1 m m

=

22

12

=

()2224

22 22 2

===

×