NEGATIVE INDEX LAW

Simplify the following and give answer in simplest rational form :

Problem 1 :

4^-1

Solution :

4^-1 = 1/4

Problem 2 :

9^-2

Solution :

9^-2 = 1/9²

Here, 9² = 9 x 9

= 1/(9 x 9)

= 1/81

Problem 3 :

3^-3

Solution :

3^-3 = 1/3³

Here, 3³ = 3 x 3 x 3

= 1/(3 x 3 x 3)

= 1/27

Problem 4 :

10^-5

Solution :

10^-5 = 1/10⁵

Here, 10⁵ = 10 x 10 x 10 x 10 x 10

= 100000

So, the answer is

= 1/100000

Problem 5 :

(1/2)^-1

Solution :

(1/2)^-1

Here the base is fraction, when we take the reciprocal of base, we get 2/1

(1/2)^-1 = (2/1)¹

Now distributing the power for numerator and denominator, we get

= 2/1

So, the answer is 2.

Problem 6 :

(2/3)^-2

Solution :

(2/3)^-2

Here the base is fraction, when we take the reciprocal of base, we get 3/2

(2/3)^-2 = (3/2)²

Distributing the power for both numerator and denominator, we get

= 9/4

Problem 7 :

(1  3/4)^-2

Solution :

(1  3/4)^-2

Converting the mixed fraction into improper fraction, we get

1  3/4 = 7/4 

(1  3/4)^-2 = (7/4)^-2

= (4/7)²

Distributing the power, we get

= 16/49

Problem 8 :

2⁰ + 2^-1

Solution :

2⁰ + 2^-1

= 1 + (1/2)

= 3/2

Problem 8 :

2⁰ + 2^-1

Solution :

2⁰ + 2^-1

= 1 + (1/2)

= 3/2

Problem 9 :

3⁰ + 3¹ - 3^-1

Solution :

= 3⁰ + 3¹ - 3^-1

= 1 + 3 - (1/3)

= 4 - (1/3)

= 11/3

Problem 10 :

2a^-1

Solution :

= 2a^-1

= 2(1/a)

= 2/a

Problem 11 :

(5c)^-2

Solution :

= (5c)^-2

Converting the negative exponent as positive exponent.

= 1/(5c)²

= 1/25c²

Problem 12 :

2(ab)^-1

Solution :

= 2(ab)^-1

Converting the negative exponent as positive exponent.

= 2(1/(ab))

= 2/ab

Problem 13 :

2ab^-1

Solution :

= 2ab^-1

Converting the negative exponent as positive exponent.

= 2a(1/b)

= 2a/b

Problem 14 :

(3n^-2)^-1

Solution :

= (3n^-2)^-1

Considering the innermost term,

= [3(1/n²)]^-1

= [3/n²]^-1

= n² / 3