SIMPLIFYING EXPRESSIONS WITH NEGATIVE EXPONENTS

To understand negative exponents, first let us see quotient rule.

To get rid of the negative exponent, we have to flip the base.

Evaluate the following without using calculator.

Example 1 :

5^-2

Solution :

To change the negative exponent, we will flip the base. Here 5 is the base, it is a integer. We can consider 5 as 5/1.

When we flip 5/1, we get 1/5.

Example 2 :

25^-1/2

Solution :

To change the negative exponent, we will flip the base.

Example 3 :

(16/25)^-3/2

Solution :

Here the base is a fraction, to get rid of the negative exponent, we will flip the base.

The reciprocal of 16/25 is 25/16.

Example 4 :

(4y)^-2

Solution :

Example 5 :

(27b)^1/3 / (9b)^-1/2

Solution :

= (27b)^1/3 / (9b)^-1/2

= (27b)^1/3 ⋅ (9b)^1/2

27 = 3 ⋅ 3 ⋅ 3 ==>  3³  and 9 = 3 ⋅ 3 ==> 3²

= (3³b)^1/3 ⋅ (3²b)^1/2

= 3b^1/3 ⋅ 3b^1/2

=  9 b^{(1/3 + 1/2)}

= 9 b^5/6

Example 6 :

b^-1/2 = 4, what is the value of b ?

Solution :

b^-1/2 = 4

(1/b)^1/2 = 4

To get rid of the exponent 1/2, we take squares on both sides.

((1/b)^1/2)² = 4²

1/b = 16

Take the reciprocal on both sides, we get

b = 1/16

Example 7 :

(2^m)^-6 = 16, what is the value of 2³ x m ?

Solution :

(2^m)^-6 = 16

Here 2^m is the base, when we flip it

2^-6m = 2⁴

Since the bases are equal, we can equate the powers.

-6m = 4

m = -2/3

2³ x m = 8 x (2/3)

= 16/3

Example 8 :

If 4ⁿ = 20, then what is the value of 4^-n ?

Solution :

4ⁿ = 20

Take the reciprocal on both sides

1/4ⁿ = 1/20

4^-n = 1/20

Example 9 :

If a^-1/2 = 3, what is the value of a ?

Solution :

a^-1/2 = 3

Taking squares on both sides, we get

(a^-1/2)² = 3²

a-1  =  9

1/a = 9

a = 1/9

Example 10 :

If 3^x = 10, what is the value of 3^x-3 ?

Solution :

Given :

3^x = 10

3^x-3  = 3^x ⋅ 3^-3

= 10 ⋅ 3^-3

= 10/3³

= 10/27