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.

● Exponents

Exponents

Laws of Exponents

Rational Exponent

Integral Exponents of a Rational Numbers

Solved Examples on Exponents

Practice Test on Exponents

● Exponents - Worksheets

Worksheet on Exponents

8th Grade Math Practice

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