# Integral Exponents

In this section, we will discuss positive and negative integral exponents of rational numbers.

Positive Integral Exponents of a Rational Number

Let a/b any rational number and positive exponent to it be 'n' then

(a/b)n = a/b x a/b x a/b x ....x n times

It means, (a x a x ...x n times)/(b x b x b x ...x n times )

In short it can written as,

(a/b)n = an/bn

Examples :

1) (3/2)3

Solution :
(3/2)3

= 33/23

= (3 x 3 x 3)/(2 x 2 x 2 )

= 27/8

2) (5/-3)4

Solution :
(5/-3)4

= 54/(-3)4

= (5 x 5 x 5 x 5)/[(-3) x (-3) x (-3) x (-3)]

= 625/81

3) [(-2)/7]3

Solution :
[(-2)/7]3

= (-2)3/73

=[(-2) x (-2) x (-2)]/(7 x 7 x 7)

= -8/343

4) (5/11)0

Solution :
(5/11)0

= 50/110

= 1/1

= 1

Negative Integral of a Rational Number

Let a/b any rational number and negative exponent to it be 'n' then

(a/b) -n = 1 ÷ [a/b x a/b x a/b x ....x n times]

It means, 1 ÷ [(a x a x ...x n times)/(b x b x b x ...x n times )]

So, (a/b)-n = (b x b x b x ...x n times)/(a x a x ...x n times)
[ As there is division so flip the numbers so a/b ----> b/a]

(a/b) -n = (b/a)n = bn/an

Examples :

1) (3/5)-2

Solution :
(3/5)-2

As the exponent is negative, so flip the numbers.

3/5 ----> 5/3 with positive exponent 2

(3/5)-2 = (5/3)2

(5/3)2

= 52/32

= 25/9

2) [(-7)/2]-3

Solution :
[(-7)/2]-3

As the exponent is negative, so flip the numbers.

(-7)/2 ----> 2/(-7) with positive exponent 3

[2/(-7)]3

= 23/(-7)3

= 8/-343
Exponents

Laws of Exponents
Rational Exponents
Integral Exponents
Scientific notation
Solved examples on Scientific Notation
Solved Examples on Exponents