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

Write each of the followings as fractions

Problem 14 :

8^-2/3

Solution :

= 8^-2/3

Writing 8 in exponential form, we get

8 = 2 • 2 • 2

= 2³

8^-2/3 = (2³)^-2/3

= 2^3(-2/3)

= 2^-2

= 1/2²

= 1/4

Problem 15 :

25^-3/2

Solution :

= 25^-3/2

Writing 25 in exponential form, we get

25 = 5 • 5

= 5²

25^-3/2 = (5²)^-3/2

= 5^{3 • (-2/3)}

= 5^-2

= 1/5²

= 1/25

Problem 16 :

Arrange in order from smallest to largest

1/50, 5^-2, 3/10, 2^-3

Solution : 

Arranging from smallest to greatest, 

0.04, 0.05, 0.3, 0.125

Arranging in exponential form, we get

5^-2, 1/50, 3/10, 2^-3

Problem 17 :

Simplify : 2^-2 + 3^-2 x 2^-3

Solution : 

= 2^-2 + 3^-2 x 2^-3

Converting each negative exponent to positive exponent, we get

= 1/2²  + 1/3² x 1/2³

= 1/4 + 1/9 x 1/8

Using order of operation, we have to perform multiplication first.

= 1/4 + 1/72

Least common multiple of 4 and 72 is 72

= (1/4) x (18/18) + 1/72

= 18/72 + 1/72

= (18 + 1)/72

= 19/72

Problem 18 :

Given that 

2^m + 2ⁿ = 9/32

Work out mn

Solution : 

2^m + 2ⁿ = 9/32

2^m + 2ⁿ = (1 + 8)/32

Decomposing into two fractions, we get

2^m + 2ⁿ = 1/32 + 8/32

= 1/32 + 1/4

= 1/2⁵ + 1/2²

2^m + 2ⁿ  = 2^-5 + 2^-2

Comparing the exponents, we get

m = -5 and n = -2

mn = -5(-2)

= 10

Problem 19 :

Put the expressions above in order, from smallest to largest,

when: x = 2 and x = 0.5

x^-2, x⁰, x, x³

Solution :

Given that, x^-2, x⁰, x, x³

Changing the negative exponents into positive exponents, we get

= 1/x², x⁰, x, x³

By applying x = 2, we get

= 1/2², 2⁰, 2, 2³

= 1/4, 1, 2, 8

It is in the order from least to greatest.

By applying x = 0.5, we get

= 1/0.5², 0.5⁰, 0.5, 0.5³

= 1/0.25, 1, 0.5, 0.125

= 4, 1, 0.5, 0.125

Arranging from least to greatest, we get

= 0.125, 0.5, 1, 4

Problem 20 :

What is the value of the missing exponent in the equation (3x^?y⁴)^-3 = x⁶/27y¹² ?

a) 2    b)  -2   c)  3   d) -3

Solution :

(3x^?y⁴)^-3 = x⁶/27y¹²

Converting the negative exponent to positive, we get

1/(3x^?y⁴)³ = x⁶/27y¹²

1/(3³(x^?)³(y⁴)³) = x⁶/27y¹²

1/(27(x^?)³(y¹²) = x⁶/27y¹²

Let a be the unknown

x^-3a = x⁶

-3a = 6

a = -6/3

a = -2

So, the missing digit is -2.

Problem 21 :

Which expression is equivalent to (-3 • 6⁰ • 4)^-2 ?

a) -144    b)  144   c)  1/144   d) -1/144

Solution :

= (-3 • 6⁰ • 4)^-2

Anything to the power 0 is 1.

= (-3 • 4)^-2

= 1/(-12)²

= 1/144

So, option c is correct.