Cambridge Additional Mathematics

108 Surds, indices, and exponentials (Chapter 4)

Historical note

1=1^3

3+5=8=2^3

7+9+11=

..

.

27 = 3^3

Nicomachusdiscovered an interesting number pattern involving cubes

and sums of odd numbers.

Theindex lawsfor m,n 2 Z are:

am£an=am+n Tomultiplynumbers with thesame base, keep the base

andaddthe indices.

am

an

=am¡n, a 6 =0 Todividenumbers with the same base, keep the base and

subtractthe indices.

(am)n=am£n Whenraisingapowerto apower, keep the base and

multiplythe indices.

(ab)n=anbn The power of a product is the product of the powers.

³

a

b

́n

=

an

bn

, b 6 =0 The power of a quotient is the quotient of the powers.

a^0 =1, a 6 =0 Any non-zero number raised to the power of zero is 1.

a¡n=

1

an

and

1

a¡n

=an and in particular a¡^1 =

1

a

, a 6 =0.

Example 9 Self Tutor

Simplify using the index laws:

a 35 £ 34 b

53

55

c

¡

m^4

¢ 3

a 35 £ 34

=35+4

=3^9

b

53

55

=5^3 ¡^5

=5¡^2

= 251

c

¡

m^4

¢ 3

=m^4 £^3

=m^12

EXERCISE 4C

1 Simplify using the index laws:

a 54 £ 57 b d^2 £d^6 c

k^8

k^3

d

75

76

e

¡

x^2

¢ 5

f

¡

34

¢ 4

g

p^3

p^7

h n^3 £n^9 i (5t)

3

j 7 x£ 72 k

103

10 q

l

¡

c^4

¢m

C Index laws

Nicomachus was born in Roman Syria (now Jerash, Jordan) around

100 AD. He wrote in Greek, and was a Pythagorean, which means he

followed the teaching ofPythagoras.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_04\108CamAdd_04.cdr Thursday, 3 April 2014 5:10:38 PM BRIAN