InChapter 6we saw the followingexponent lawswhich are true for all positive basesaandband all

integer indicesmandn.

² am£an=am+n Tomultiplynumbers with the same base, keep the base andadd

the indices.

²

am

an

=am¡n Todividenumbers with the same base, keep the base andsubtract

the indices.

² (am)n=am£n When raising a power to a power, keep the base andmultiplythe

indices.

² (ab)n=anbn The power of a product is the product of the powers.

²

³a

b

́n

=

an

bn

The power of a quotient is the quotient of the powers.

² a^0 =1, a 6 =0 Any non-zero number raised to the power of zero is 1.

² a¡n=

1

an

and

1

a¡n

=an and in particular a¡^1 =

1

a

.

These laws can also be applied to rational exponents, or exponents which are written as a fraction.

We have seen examples of rational indices already when we studied surds.

a

(^12)

p
a.
and (a
(^13)
)^3 =a
(^13) £ 3
=a^1 =a and (^3
p
a)^3 =a,soa
(^13)
=^3
p
a.
In general, a
(^1) n
= n
p
a wheren
p
ais called the ‘nth root ofa’.
Example 1 Self Tutor
Simplify: a 49
(^12)
b 27
(^13)
c 49
¡^12
d 27
¡^13
a 49
(^12)

=

p

49

=7

b 27

(^13)

=^3

p

27

=3

c 49

¡^12

=

1

49

(^12)

=

1

p

49

=^17

d 27 ¡

(^13)

=

1

27

(^13)

=

1

p (^327)
=^13
Discovery 1 Rational Exponents
#endboxedheading
Our aim is to discover the meaning of numbers raised to rational exponents of the form mn where
A RATIONAL EXPONENTS [1.9, 2.4]
Notice that (a
(^12)
)^2 =a
(^12) £ 2
=a^1 =a and (
p
a)^2 =a,so
m,n 2 Z. For example, what does 8
(^23)
mean?
566 Exponential functions and equations (Chapter 28)
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Y:\HAESE\IGCSE01\IG01_28\566IGCSE01_28.CDR Monday, 27 October 2008 2:43:33 PM PETER