# 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}= a^{n}/b^{n}**Examples :**

1) (3/2)

^{3}

**Solution :**

(3/2)

^{3}

= 3

^{3}/2

^{3}

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

= 27/8

2) (5/-3)

^{4}

**Solution :**

(5/-3)

^{4}

= 5

^{4}/(-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}/7

^{3}

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

= -8/343

4) (5/11)

^{0}

**Solution :**

(5/11)

^{0}

= 5

^{0}/11

^{0}

= 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}= b^{n}/a^{n}**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}

= 5

^{2}/3

^{2}

= 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}

= 2

^{3}/(-7)

^{3}

= 8/-343

**Exponents**

• Laws of Exponents

• Rational Exponents

• Integral Exponents

• Scientific notation

• Solved examples on Scientific Notation

• Solved Examples on Exponents

• Laws of Exponents

• Rational Exponents

• Integral Exponents

• Scientific notation

• Solved examples on Scientific Notation

• Solved Examples on Exponents