SIMPLIFYING EXPRESSIONS WITH NEGATIVE EXPONENTS WORKSHEET

Evaluate the following without using calculator.

(1) 5^-2                   Solution

(2) 25^-1/2                  Solution

(3) (16/25)^-3/2                Solution

(4) (4y)^-2             Solution

(5) (27b)^1/3 / (9b)^-1/2             Solution

(6) b^-1/2 = 4, what is the value of b ?              Solution

(7) (2^m)^-6 = 16, what is the value of 2³ x m ?       Solution

(8) If 4ⁿ = 20, then what is the value of 4^-n ?         Solution

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

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

(11)  A drop of water leaks from a faucet every second. How many liters of water leak from the faucet in 1 hour?

Solution

(12)

a)  If a is a nonzero number, does the value of a⁰ depend on the value of a?

b)  Explain how to evaluate 10⁻³

c)  Without evaluating, order 5⁰, 5⁴, and 5⁻⁵ from least to greatest.

Solution

(13) 

Which is different? Find “both” answers.

a)  Rewrite 1/(3 ⋅ 3 ⋅ 3)   using a negative exponent.

b)  Write 3 to the negative third power.

c)  Write 1/3 cubed as a power.

d)  Write (−3) ⋅ (−3) ⋅ (−3) as a power.

Solution

(14)

Use the Quotient of Powers Property to copy and complete the table.

a) What patterns do you see?

b)  How would you define 5⁰? Why?

c) How can you rewrite 5⁻¹ as a fraction?

Solution

Example 13 :

Describe and correct the error in evaluating the expression.

Solution

Write the expression with only positive exponents. Assume all variables represent non zero numbers. Simplify if necessary.

Problem 1 : 

(3x²)³/x¹⁵

(a) 27/x²¹  (b)  27/x¹⁰  (c)  3/x⁹  (d)  27/x⁹

Solution

Problem 2 : 

(-a)^-18

(a) 1/a¹⁸  (b)  18a  (c)  1/-a¹⁸  (d)  1/a^-18

Solution

Problem 3 : 

x^-16/x^-4

(a) 1/x¹²  (b) x¹²  (c)  1/x²⁰  (d)  -x²⁰

Solution

Problem 4 : 

(-3w³/x)⁴

(a) -81w¹²/x⁴          (b) -81w¹²/x        (c)  81w¹²/x⁴          (d)  81w⁷/x⁴

Solution

Problem 5 : 

m^-9 m⁵ m^-1

(a) 1/m⁵    (b) 1/m⁴   (c)  m⁷   (d)  m⁵

Solution

Problem 6 : 

(2^-2 ⋅ 5^-5)^-4

(a) 2⁸ ⋅ 5²⁰    (b) 1/(2⁷ ⋅ 5⁷)  (c)  1/(2⁸ ⋅ 5²⁰)   (d)  2⁷ ⋅ 5⁷

Solution

Problem 7 : 

By what number should (–3)^{– 2} be multiplied so that the product may be equal to 9?

Solution

Problem 8 : 

Find the value of x so that

(5/3)^-2 ⋅ (5/3)^-14 = (5/3)^8x

Solution

Problem 9 : 

If 2^{2x – 3} = (64)^x, find the value of x.

Solution

Problem 10 : 

Find the value of x so that

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

Solution

Problem 11 : 

Find the value of x so that

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

Solution

Problem 12 : 

If p/q = (5/2)^-2 x (4/3)⁰ , find the value of (p/q)^-2

Solution

Simplify. Your answer should contain only positive exponents.

Problem 1 :

(x^-2 x^-3)⁴

Solution

Problem 2 :

(x⁴)^-3 ⋅ 2x⁴

Solution

Problem 3 :

(n³)³ ⋅ 2n^-1

Solution

Problem 4 :

(2v)²  ⋅ 2v²

Solution

Problem 5 :

(2x² y  · 4x² y⁴ · 3x)/3x^-3 y²

Solution

Problem 6 :

(2 y³· 3 x y³) / 3 x² y⁴

Solution

Problem 7 :

(x³· y³· x³) / 4 x²

Solution

Problem 8 :

(3 x²· y²) / (2 x^-1 · 4 yx²)

Solution

Problem 9 :

x / (2 x⁰)²

Solution

Problem 10 :

2m^-4 / (2m^-4)³

Solution

Problem 11 :

(2m²)^-1 / m²

Solution

Problem 12 :

(2x³)/(x^-1)³

Solution

Problem 13 :

(a^-3 b^-3)⁰

Solution

Problem 14 :

x⁴ y³ ⋅ (2y²)⁰

Solution

Problem 15 :

ba⁴ ⋅ (2ba⁴)^-3

Solution

Problem 16 :

(2x⁰y²)^-3 ⋅ 2yx³

Solution

Problem 17 :

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

Solution

Problem 18 :

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

Solution

Problem 19 :

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

Solution

Problem 20 :

By what number should (-512)^-1 be divided so that the quotient may be equal to 

8^-2 ?

Solution

Problem 21 :

By what number should (144/225)^-1 be divided so that the quotient may be equal to 

(12/15)^-4 ?

Solution

Problem 22 :

Simplify 6^-2 + (3/2)^-2

Solution

Problem 23 :

Find the value of x if [(3/7)³]^-2 = (3/7)^2x

Solution