EXPONENTS WITH NEGATIVE BASES WORKSHEET

Problem 1

Evaluate :

-8²

Problem 2 :

Evaluate :

(-8)²

Problem 3 :

Evaluate :

(-8)³

Problem 4 :

Evaluate :

3^-2

Problem 5 :

Evaluate :

(-3/2)^-2

Problem 6 :

Evaluate :

(-5/4)^-3

Problem 7 :

Evaluate :

(-5^-2) ⋅ (-3^-4)

Problem 8 :

Evaluate :

(-3)^-4/ (-2)^-3

Problem 9 :

It is given that -a^-1/3  =  3. Find the value of a. 

Problem 10 :

It is given that (-a)^-1/3  =  4/3. Find the value of a. 

Problem 11 :

Express (15³)^–16 as a single exponent of 15.

Problem 12 :

By what number should (7)^–2 be multiplied so that the product may be equal to (-343)^-1?

Problem 13 :

By what number should (−3/4)⁵ be multiplied so that the product may be equal to (−64/27)^-1 ?

Problem 14 :

(-7/11)^-3 x (-7/11)^5x  = [(-7/11)^-2]^-1

Problem 15 :

(3/7)^{-2x + 1} ÷ (3/7)^-1  = [(3/7)^-1]^-7

Problem 16 :

(-4/5)³ x 5² x (-1/2)⁵ x (1/2)^-3

Problem 17 :

The value of the expression (1/3)³ x (-2/5)² x (-3/2)³

Problem 18 :

The value of the expression (-1/4)^-3 x (-1/4)^-2

Problem 19 :

Find m so that (–3)^{m +1} × (–3)⁵ = (–3)⁷

Problem 20 :

If 3^{x + y} = 81 and 81^{x- y} = 3, then find the values of x and y

Detailed Answer Key

Problem 1 :

Evaluate :

-8²

Solution :

= -8²

= - 8 ⋅ 8

= - 64

Problem 2 :

Evaluate :

(-8)²

Solution :

= (-8)(-8)

= 64

Problem 3 :

Evaluate :

(-8)³

Solution :

= (-8)(-8)(-8)

= -512

Problem 4 :

Evaluate :

3^-2

Solution :

= 3^-2

= (1/3)²

Distribute the exponent to numerator and denominator.

= 1² / 3²

= 1/9

Problem 5 :

Evaluate :

(-3/2)^-2

Solution :

= (-3/2)^-2

= (-2/3)²

Since the exponent is even, the negative sign inside the parentheses will become positive.

= (2/3)²

Distribute the exponent to numerator and denominator.

= 2²/3²

= 4/9

Problem 6 :

Evaluate :

(-5/4)^-3

Solution :

= (-5/4)^-3

= (-4/5)³

Since the exponent is odd, the negative sign inside the parentheses will remain same. 

Distribute the exponent to numerator and denominator. 

= -4³/5³

= -64/125

Problem 7 :

Evaluate :

(-5^-2) ⋅ (-3^-4)

Solution :

= (-5^-2) ⋅ (-3^-4)

= (-1/5)² ⋅ (-1/3)⁴

= (1/25) ⋅ (1/81)

= 1/2025

Problem 8 :

Evaluate :

(-3)^-4/(-2)^-3

Solution :

= (-3)^-4/(-2)^-3

= (-1/3)⁴/(-1/2)³

= (1/81)/(-1/8)

= (1/81) ⋅ (-8/1)

= -8/81

Problem 9 :

It is given that -a^-1/3  =  3. Find the value of a.

Solution :

-a^1/3 = 3

Multiply each side by -1.

a^1/3 = -3

a = (-3)^3/1

= (-3)³

= (-3)(-3)(-3)

= -27

Problem 10 :

It is given that (-a)^-1/3  =  4/3. Find the value of a.

Solution :

(-a)^-1/3 = 4/3

-a = (4/3)^-3/1

-a = (4/3)^-3

-a = (3/4)³

Distribute the exponent to numerator and denominator.

-a = 3³/4³

-a = 27/64

Multiply each side by -1.

a = - 27/64

Problem 11 :

Express (15³)^–16 as a single exponent of 15.

Solution :

= (15³)^–16

Since we have power raised by another power, we can multiply the powers.

= 15^3(–16)

= 15^–48

= (1/15)⁴⁸

Problem 12 :

By what number should (7)^–2 be multiplied so that the product may be equal to (-343)^-1?

Solution :

Let x be the required number to be multiplied.

(7)^–2 ⋅ x = (-343)^-1

Converting the negative exponents as positive exponents, we get

343 = 7 ⋅ 7 ⋅ 7

= 7³

(1/7²) ⋅ x = (-7³)^-1

(1/7²) ⋅ x = (-7)^-3

(1/7²) ⋅ x = (-1/7³)

x = (-1/7³) ⋅ (7²/1)

x = -1/7

Problem 13 :

By what number should (−3/4)⁵ be multiplied so that the product may be equal to (−64/27)^-1 ?

Solution :

Let x be the required number.

(−3/4)⁵  ⋅ x = (−64/27)^-1

(−3/4)⁵  ⋅ x = (−27/64)

x = (−27/64)(−4/3)⁵

x = (−3³/4³)(−4⁵/3⁵)

= 4² / 3²

= 16/9

Problem 14 :

(-7/11)^-3 x (-7/11)^5x  = [(-7/11)^-2]^-1

Solution :

(-7/11)^-3 x (-7/11)^5x = [(-7/11)^-2]^-1

(-7/11)^-3+5x = (-7/11)²

-3 + 5x = 2

5x = 2 + 3

5x = 5

x = 5/5

x = 1

Problem 15 :

(3/7)^{-2x + 1} ÷ (3/7)^-1  = [(3/7)^-1]^-7

Solution :

(3/7)^{-2x + 1} ÷ (3/7)^-1  = [(3/7)^-1]^-7

(3/7)^{-2x + 1} x (3/7)¹  = (3/7)⁷

(3/7)^{-2x + 1 + 1} = (3/7)⁷

(3/7)^{-2x + 2} = (3/7)⁷

Since the bases are equal, by equating the powers, we get

-2x + 2 = 7

-2x = 7 - 2

-2x = 5

x = -5/2

Problem 16 :

(-4/5)³ x 5² x (-1/2)⁵ x (1/2)^-3

Solution :

= (-4/5)³ x 5² x (-1/2)⁵ x (1/2)^-3

= (-64/125) x 25 x (-1/32) x (2)³

= (-64/5) x (-1/32) x 8

= 16/5

Problem 17 :

The value of the expression (1/3)³ x (-2/5)² x (-3/2)³

Solution :

= (1/3)³ x (-2/5)² x (-3/2)³

Using the exponents, we get

= (1/27) x (4/25) x (-27/8)

= 1/50

Problem 18 :

The value of the expression (-1/4)^-3 x (-1/4)^-2

Solution :

= (-1/4)^-3 x (-1/4)^-2

To convert the negative exponents as positive exponent, we have to write each fractions as its reciprocal

= (-4)³ x (-4)²

= -64 x 16

= -1024

Problem 19 :

Find m so that (–3)^{m +1} × (–3)⁵ = (–3)⁷

Solution :

(–3)^{m +1} × (–3)⁵ = (–3)⁷

(–3)^{m + 1 + 5} = (–3)⁷

(–3)^{m + 6} = (–3)⁷

Equating the powers, we get

m + 6 = 7

m = 7 - 6

m = 1

So, the value of m is 1.

Problem 20 :

If 3^{x + y} = 81 and 81^{x- y} = 3, then find the values of x and y

Solution :

3^{x + y} = 81 and 81^{x- y} = 3

3^{x + y} = 3⁴ and 3^{4(x - y)} = 3

x + y = 4 -----(1)

4(x - y) = 3  -----(2)

From (1), y = 4 - x

4(x - (4 - x)) = 3

4(x - 4 + x) = 3

4(2x - 4) = 3

2x - 4 = 3/4

2x = (3/4) + 4 

2x = 19/4

x = 19/8

So, the value of x is 19/8

y = 4 - (19/8)

y = (32 - 19)/8

= 13/8

So, the value of y is 13/8

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