Algebra for College Students

Mark Dugopolski

Chapter 5

Exponents and Polynomials - all with Video Answers

Educators

Section 1

Integral Exponents and Scientific Notation

Problem 1

After reading this section, write out the answers to these questions. Use complete sentences.
What is an exponential expression?

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Problem 2

After reading this section, write out the answers to these questions. Use complete sentences.
What is the meaning of a negative exponent?

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Problem 3

After reading this section, write out the answers to these questions. Use complete sentences.
What is the product rule?

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Problem 4

After reading this section, write out the answers to these questions. Use complete sentences.
What is the quotient rule?

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Problem 5

After reading this section, write out the answers to these questions. Use complete sentences.
How do you convert a number from scientific notation to standard notation?

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Problem 6

After reading this section, write out the answers to these questions. Use complete sentences.
How do you convert a number from standard notation to scientific notation?

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Problem 7

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$2^{2},-2^{2},(-2)^{2}, 2^{-2},-2^{-2},(-2)^{-2}$$

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Problem 8

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$3^{2},-3^{2},(-3)^{2}, 3^{-2},-3^{-2},(-3)^{-2}$$

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Problem 9

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$2^{3},-2^{3},(-2)^{3}, 2^{-3},-2^{-3},(-2)^{-3}$$

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Problem 10

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$4^{3},-4^{3},(-4)^{3}, 4^{-3},-4^{-3},(-4)^{-3}$$

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Problem 11

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\frac{1}{5^{2}}, \frac{1}{-5^{2}}, \frac{1}{(-5)^{2}}, \frac{1}{5^{-2}}, \frac{1}{-5^{-2}}, \frac{1}{(-5)^{-2}}$$

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Problem 12

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\frac{1}{4^{2}}, \frac{1}{-4^{2}}, \frac{1}{(-4)^{2}}, \frac{1}{4^{-2}}, \frac{1}{-4^{-2}}, \frac{1}{(-4)^{-2}}$$

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Problem 13

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$7^{1},-7^{1},(-7)^{1}, 7^{-1},-7^{-1},(-7)^{-1}$$

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Problem 14

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\left(\frac{1}{6}\right)^{1},\left(-\frac{1}{6}\right)^{1},-\left(\frac{1}{6}\right)^{1},\left(\frac{1}{6}\right)^{-1},\left(-\frac{1}{6}\right)^{-1},-\left(\frac{1}{6}\right)^{-1}$$

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Problem 15

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\left(\frac{1}{3}\right)^{2},\left(-\frac{1}{3}\right)^{2},-\left(\frac{1}{3}\right)^{2},\left(\frac{1}{3}\right)^{-2},\left(-\frac{1}{3}\right)^{-2},-\left(\frac{1}{3}\right)^{-2}$$

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Problem 16

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\left(\frac{1}{3}\right)^{2},\left(-\frac{1}{3}\right)^{2},-\left(\frac{1}{3}\right)^{2},\left(\frac{1}{3}\right)^{-2},\left(-\frac{1}{3}\right)^{-2},-\left(\frac{1}{3}\right)^{-2}$$

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Problem 17

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\left(\frac{2}{3}\right)^{3},\left(-\frac{2}{3}\right)^{3},-\left(\frac{2}{3}\right)^{3},\left(\frac{2}{3}\right)^{-3},\left(-\frac{2}{3}\right)^{-3},-\left(\frac{2}{3}\right)^{-3}$$

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Problem 18

Each of Exercises 7–18 contains six similar expressions. For each exercise evaluate the six expressions and note their similarities and differences. See Example 1.
$$\left(\frac{3}{5}\right)^{3},\left(-\frac{3}{5}\right)^{3},-\left(\frac{3}{5}\right)^{3},\left(\frac{3}{5}\right)^{-3},\left(-\frac{3}{5}\right)^{-3},-\left(\frac{3}{5}\right)^{-3}$$

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Problem 19

For all exercises in this section, assume that the variables represent nonzero real numbers and use only positive exponents in your answers. Simplify. See Example 2.
$$2^{5} \cdot 2^{12}$$

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Problem 20

For all exercises in this section, assume that the variables represent nonzero real numbers and use only positive exponents in your answers. Simplify. See Example 2.
$$3^{15} \cdot 3^{-3}$$

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Problem 21

For all exercises in this section, assume that the variables represent nonzero real numbers and use only positive exponents in your answers. Simplify. See Example 2.
$$-2 x^{-7} \cdot 3 x$$

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Problem 22

For all exercises in this section, assume that the variables represent nonzero real numbers and use only positive exponents in your answers. Simplify. See Example 2.
$$-5 a \cdot 6 a^{-12}$$

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Problem 23

For all exercises in this section, assume that the variables represent nonzero real numbers and use only positive exponents in your answers. Simplify. See Example 2.
$$-7 b^{-7}\left(-3 b^{-3}\right)$$

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Problem 24

For all exercises in this section, assume that the variables represent nonzero real numbers and use only positive exponents in your answers. Simplify. See Example 2.
$$-\frac{1}{2} w^{-4} \cdot\left(-6 w^{-2}\right)$$

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Problem 25

Simplify each expression. See Example 3.
$$3^{0},-3^{0},(-3)^{0},-(-3)^{0}$$

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Problem 26

Simplify each expression. See Example 3.
$$2 a^{0},-2 a^{0},(-2 a)^{0},-2(-a)^{0}$$

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Problem 27

Simplify each expression. See Example 3.
$$(2+3)^{0}, 2^{0}+3^{0},\left(2^{0}+3\right)^{0}$$

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Problem 28

Simplify each expression. See Example 3.
$$(4-9)^{0}, 4^{0}-9^{0}, 4^{0}-9$$

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Problem 29

Simplify each expression. See Example 3.
$$3 s t^{0}, 3(s t)^{0},(3 s t)^{0}$$

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Problem 30

Simplify each expression. See Example 3.
$$-4 x y^{0},-4 x^{0} y,-(4 x y)^{0}$$

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Problem 31

Simplify each expression. See Example 3.
$$2 w^{-3}\left(w^{7} \cdot w^{-4}\right)$$

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Problem 32

Simplify each expression. See Example 3.
$$5 y^{2} z\left(y^{-3} z^{-1}\right)$$

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Problem 33

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{2}{4^{-2}}$$

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Problem 34

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{5}{10^{-3}}$$

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Problem 35

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{3^{-1}}{10^{-2}}$$

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Problem 36

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{2 y^{-2}}{3^{-1}}$$

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Problem 37

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{2 x^{-3}(4 x)}{5 y^{-2}}$$

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Problem 38

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{5^{-2} x y^{-3}}{3 x^{-2}}$$

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Problem 39

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{4^{-2} x^{3} x^{-6}}{3 x^{-3} x^{2}}$$

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Problem 40

Write each expression without negative exponents and simplify. See Example 4.
$$\frac{3 y^{-4} y^{-6}}{2^{-3} y^{2} y^{-7}}$$

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Problem 41

Simplify each expression. See Examples 5 and 6.
$$\frac{x^{5}}{x^{3}}$$

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Problem 42

Simplify each expression. See Examples 5 and 6.
$$\frac{a^{8}}{a^{3}}$$

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Problem 43

Simplify each expression. See Examples 5 and 6.
$$\frac{3^{6}}{3^{-2}}$$

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Problem 44

Simplify each expression. See Examples 5 and 6.
$$\frac{6^{2}}{6^{-5}}$$

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Problem 45

Simplify each expression. See Examples 5 and 6.
$$\frac{4 a^{-5}}{12 a^{-2}}$$

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Problem 46

Simplify each expression. See Examples 5 and 6.
$$\frac{-3 a^{-3}}{-21 a^{-4}}$$

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Problem 47

Simplify each expression. See Examples 5 and 6.
$$\frac{-614^{-5}}{2 w^{3}}$$

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Problem 48

Simplify each expression. See Examples 5 and 6.
$$\frac{10 x^{-6}}{-2 x^{2}}$$

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Problem 49

Simplify each expression. See Examples 5 and 6.
$$\frac{3^{3} w^{-2} w^{5}}{3^{-5} w^{-3}}$$

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Problem 50

Simplify each expression. See Examples 5 and 6.
$$\frac{2^{-3} w^{5}}{2^{5} w^{3} w^{-7}}$$

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Problem 51

Simplify each expression. See Examples 5 and 6.
$$\frac{3 x^{-6} \cdot x^{2} y^{-1}}{6 x^{-5} y^{-2}}$$

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Problem 52

Simplify each expression. See Examples 5 and 6.
$$\frac{2 r^{-3} t^{-1}}{10 r^{5} t^{2} \cdot t^{-3}}$$

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Problem 53

Use the rules of exponents to simplify each expression.
$$3^{-1}\left(\frac{1}{3}\right)^{-3}$$

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Problem 54

Use the rules of exponents to simplify each expression.
$$2^{-2}\left(\frac{1}{4}\right)^{-3}$$

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Problem 55

Use the rules of exponents to simplify each expression.
$$-2^{4}+\left(\frac{1}{2}\right)^{-1}$$

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Problem 56

Use the rules of exponents to simplify each expression.
$$-3^{4}-(-3)^{4}$$

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Problem 57

Use the rules of exponents to simplify each expression.
$$-(-2)^{-3} \cdot 2^{-1}$$

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Problem 58

Use the rules of exponents to simplify each expression.
$$-(-3)^{-1} \cdot 9^{-1}$$

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Problem 59

Use the rules of exponents to simplify each expression.
$$7 \cdot 2^{-3}-2 \cdot 4^{-1}$$

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Problem 60

Use the rules of exponents to simplify each expression.
$$5 \cdot 3^{-2}-2 \cdot 5^{0} \cdot 3^{-1}$$

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Problem 61

Use the rules of exponents to simplify each expression.
$$\left(1+2^{-1}\right)^{-2}$$

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Problem 62

Use the rules of exponents to simplify each expression.
$$\left(2^{-1}+2^{-1}\right)^{-2}$$

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Problem 63

Use the rules of exponents to simplify each expression.
$$2 x^{2} \cdot 5 x^{-5}$$

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Problem 64

Use the rules of exponents to simplify each expression.
$$2 x^{2} \cdot 5 y^{-5}$$

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Problem 65

Use the rules of exponents to simplify each expression.
$$\frac{-3 a^{5}\left(-2 a^{-1}\right)}{6 a^{3}}$$

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Problem 66

Use the rules of exponents to simplify each expression.
$$\frac{6 a\left(-a b^{-2}\right)}{-2 a^{2} b^{-3}}$$

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Problem 67

Use the rules of exponents to simplify each expression.
$$\frac{\left(-3 x^{3} y^{2}\right)\left(-2 x y^{-3}\right)}{-9 x^{2} y^{-5}}$$

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Problem 68

Use the rules of exponents to simplify each expression.
$$\frac{\left(-2 x^{-5} y\right)\left(-3 x y^{6}\right)}{-6 x^{-6} y^{2}}$$

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Problem 69

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$8=2^{?}$$

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Problem 70

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$27=3^{?}$$

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Problem 71

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$\frac{1}{4}=2^{?}$$

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Problem 72

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$\frac{1}{125}=5^{?}$$

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Problem 73

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$16=\left(\frac{1}{2}\right)^{?}$$

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Problem 74

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$81=\left(\frac{1}{3}\right)^{2}$$

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Problem 75

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$10^{?}=0.001$$

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Problem 76

For each equation, find the integer that can be used as the exponent to make the equation correct.
$$10^{?}=10,000$$

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Problem 77

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$4.86 \times 10^{8}$$

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Problem 78

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$3.80 \times 10^{2}$$

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Problem 79

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$2.37 \times 10^{-6}$$

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Problem 80

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$1.62 \times 10^{-3}$$

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Problem 81

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$4 \times 10^{6}$$

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Problem 82

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$496 \times 10^{3}$$

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Problem 83

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$5 \times 10^{-6}$$

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Problem 84

Write each number in standard notation. See Example 7.
See the Strategy for Converting to Standard Notation box on page 299.
$$48 \times 10^{-3}$$

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Problem 85

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$320,000$$

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Problem 86

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$43,298,000$$

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Problem 87

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$0.00000071$$

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Problem 88

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$0.00000894$$

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Problem 89

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$0.0000703$$

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Problem 90

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$8,200,100$$

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Problem 91

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$205 \times 10^{5}$$

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Problem 92

Write each number in scientific notation. See Example 8.
See the Strategy for Converting to Scientific Notation box on page 299.
$$0.403 \times 10^{-9}$$

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Problem 93

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$( 4000)( 5000)( 0.0003)$$

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Problem 94

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$(50,000)(0.00002)(100)$$

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Problem 95

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$\frac{(5,000,000)(0.0003)}{2000}$$

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Problem 96

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$\frac{(6000)(0.00004)}{(30,000)(0.002)}$$

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Problem 97

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$\frac{6 \times 10^{40}}{2 \times 10^{18}}$$

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Problem 98

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$\frac{4.6 \times 10^{12}}{2.3 \times 10^{5}}$$

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Problem 99

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$\frac{\left(-4 \times 10^{5}\right)\left(6 \times 10^{-9}\right)}{2 \times 10^{-16}}$$

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Problem 100

Evaluate each expression using scientific notation without a calculator. See Example 9.
$$\frac{\left(4.8 \times 10^{-3}\right)\left(5 \times 10^{-8}\right)}{\left(1.2 \times 10^{-6}\right)\left(2 \times 10^{12}\right)}$$

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Problem 101

Evaluate each expression using a calculator. Write answers in scientific notation. Round the decimal part to three decimal places.
$$\left(4.3 \times 10^{9}\right)\left(3.67 \times 10^{-5}\right)$$

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Problem 102

Evaluate each expression using a calculator. Write answers in scientific notation. Round the decimal part to three decimal places.
$$\left(2.34 \times 10^{6}\right)\left(8.7 \times 10^{5}\right)$$

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Problem 103

Evaluate each expression using a calculator. Write answers in scientific notation. Round the decimal part to three decimal places.
$$\left(4.37 \times 10^{-6}\right)+\left(8.75 \times 10^{-5}\right)$$

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Problem 104

Evaluate each expression using a calculator. Write answers in scientific notation. Round the decimal part to three decimal places.
$$\left(6.72 \times 10^{5}\right)+\left(8.98 \times 10^{6}\right)$$

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Problem 105

Evaluate each expression using a calculator. Write answers in scientific notation. Round the decimal part to three decimal places.
$$\frac{\left(5.6 \times 10^{14}\right)^{2}\left(3.2 \times 10^{-6}\right)}{\left(6.4 \times 10^{-3}\right)^{3}}$$

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Problem 106

Evaluate each expression using a calculator. Write answers in scientific notation. Round the decimal part to three decimal places.
$$\frac{\left(3.51 \times 10^{-6}\right)^{3}(4000)^{5}}{2 \pi}$$

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Problem 107

Solve each problem. Round to three decimal places. See Example 10.
Distance to the sun. The distance from the earth to the sun is 93 million miles. Express this distance in feet using scientific notation ( 1 mile $=5280$ feet).

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Problem 108

Solve each problem. Round to three decimal places. See Example 10.
Traveling time. The speed of light is $9.83569 \times 10^{8}$ feet per second. How long does it take light to get from the sun to the earth? (See Exercise 107 )

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Problem 109

Solve each problem. Round to three decimal places. See Example 10.
Space travel. How long does it take a spacecraft traveling $1.2 \times 10^{5}$ kilometers per second to travel $4.6 \times 10^{12}$ kilometers?

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Problem 110

Solve each problem. Round to three decimal places. See Example 10.
Diameter of a dot. If the circumference of a very small circle is $2.35 \times 10^{-8}$ meter, then what is the diameter of the circle?

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Problem 111

Solve each problem. Round to three decimal places. See Example 10.
Solid waste per person. In 1960 the $1.80863 \times 10^{8}$ people in the United States generated $8.71 \times 10^{7}$ tons of municipal solid waste (Environmental Protection Agency, www.epa.gov). How many pounds per person per day were generated in $1960 ?$

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Problem 112

Solve each problem. Round to three decimal places. See Example 10.
An increasing problem. According to the EPA, in 2002 the $2.86843 \times 10^{8}$ people in the United States generated $4.8 \times 10^{11}$ pounds of solid municipal waste.
a) How many pounds per person per day were generated in $2002 ?$
b) Use the graph to predict the number of pounds per person per day that will be generated in the year 2010

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Problem 113

Exploration
a) Using pairs of integers, find values for $m$ and $n$ for which $2^{m} \cdot 3^{n}=6^{m+n}$
b) For which values of $m$ and $n$ is it true that $2^{m} \cdot 3^{n} \neq$ $6^{m+n} ?$

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Problem 114

Cooperative learning
Work in a group to find the units digit of $3^{99}$ and explain how you found it.

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Problem 115

Discussion
What is the difference between $-a^{n}$ and $(-a)^{n}$ where $n$ is an integer? For which values of $a$ and $n$ do they have the same value, and for which values of $a$ and $n$ do they have different values?

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Problem 116

Exploration
If $a+b=a,$ then what can you conclude about $b ?$ Use scientific notation on your calculator to find $5 \times 10^{20}+3 \times 10^{6} .$ Explain why your calculator displays the answer that it gets.

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