# 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.

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● **Exponents**

**Exponents**

**Laws of Exponents**

**Rational Exponent**

**Integral Exponents of a Rational Numbers**

**Solved Examples on Exponents**

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