Integral Exponents of a Rational Numbers

We shall be dealing with the positive and negative integral exponents of a rational numbers.

Positive Integral Exponent of a Rational Number

Let a/b be any rational number and n be a positive integer. Then,

(a/b)ⁿ = a/b × a/b × a/b × ……. n times

= (a × a × a ×…….. n times )/( b × b × b ×……….. n times )

= aⁿ/bⁿ

Thus (a/b)ⁿ = aⁿ/bⁿ for every positive integer n .

For example:

Evaluate:

(i) (3/5)³

= 3³/5³

= 3 × 3 × 3/5 × 5 × 5

= 27/125

(ii) (-3/4)⁴

= (-3)⁴/4⁴

= 34/44

= 3 × 3 × 3 × 3/4 × 4 × 4 × 4

= 81/256

(iii) (-2/3)⁵

= (-2)⁵/3⁵

= (-2)⁵/3⁵

= -2 × -2 × -2 × -2 × -2/3 × 3 × 3 × 3 × 3

= -32/243

Negative Integral Exponent of a Rational Number

Let a/b be any rational number and n be a positive integer.

Then, we define, (a/b) $^{-n}$ = (b/a)ⁿ

For example:

(i) (3/4) $^{-5}$

= (4/3)⁵

(ii) 4 $^{-6}$

= (4/1) $^{-6}$

= (1/4)⁶

Also, we define, (a/b) = 1

Evaluate:

(i) (2/3) $^{-3}$

= (3/2)³

= 3³/2³

= 27/8

(ii) 4 $^{-2}$

= (4/1) $^{-2}$

= (1/4)²

= 1²/4²

= 1/16

(iii) (1/6) $^{-2}$

= (6/1)²

= 6²

= 36

(iv) (2/3) = 1

The positive and negative integral exponents of a rational numbers are explained here with examples.