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1.3E: Exercises

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Practice Makes Perfect

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

Exercise 1.3E.55

169

Answer

16 minus 9, the difference of sixteen and nine

Exercise 1.3E.56

39

Exercise 1.3E.57

28÷4

Answer

28 divided by 4, the quotient of twenty-eight and four

Exercise 1.3E.58

x+11

Exercise 1.3E.59

(2)(7)

Answer

2 times 7, the product of two and seven

Exercise 1.3E.60

(4)(8)

Exercise 1.3E.61

14<21

Answer

fourteen is less than twenty-one

Exercise 1.3E.62

17<35

Exercise 1.3E.63

3619

Answer

thirty-six is greater than or equal to nineteen

Exercise 1.3E.64

6n=36

Exercise 1.3E.65

y1>6

Answer

y minus 1 is greater than 6, the difference of y and one is greater than six

Exercise 1.3E.66

y4>8

Exercise 1.3E.67

218÷6

Answer

2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six

Exercise 1.3E.68

a112

In the following exercises, determine if each is an expression or an equation.

Exercise 1.3E.69

96=54

Answer

equation

Exercise 1.3E.70

79=63

Exercise 1.3E.71

54+3

Answer

expression

Exercise 1.3E.72

x+7

Exercise 1.3E.73

x+9

Answer

expression

Exercise 1.3E.74

y5=25

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

Exercise 1.3E.75

53

Answer

125

Exercise 1.3E.76

83

Exercise 1.3E.77

28

Answer

256

Exercise 1.3E.78

105

In the following exercises, simplify using the order of operations.

Exercise 1.3E.79
  1. 3+85
  2. (3+8)5
Answer
  1. 43
  2. 55
Exercise 1.3E.80
  1. 2+63
  2. (2+6)3
Exercise 1.3E.81

2312÷(95)

Answer

5

Exercise 1.3E.82

3218÷(115)

Exercise 1.3E.83

38+52

Answer

34

Exercise 1.3E.84

47+35

Exercise 1.3E.85

2+8(6+1)

Answer

58

Exercise 1.3E.86

4+6(3+6)

Exercise 1.3E.87

412/8

Answer

6

Exercise 1.3E.88

236/6

Exercise 1.3E.89

(6+10)÷(2+2)

Answer

4

Exercise 1.3E.90

(9+12)÷(3+4)

Exercise 1.3E.91

20÷4+65

Answer

35

Exercise 1.3E.92

33÷3+82

Exercise 1.3E.93

32+72

Answer

58

Exercise 1.3E.94

(3+7)2

Exercise 1.3E.95

3(1+96)42

Answer

149

Exercise 1.3E.96

5(2+84)72

Exercise 1.3E.97

2[1+3(102)]

Answer

50

Exercise 1.3E.98

5[2+4(32)]

Evaluate an Expression

In the following exercises, evaluate the following expressions.

Exercise 1.3E.99

7x+8 when x=2

Answer

22

Exercise 1.3E.100

8x6 when x=7

Exercise 1.3E.101

x2 when x=12

Answer

144

Exercise 1.3E.102

x3 when x=5

Exercise 1.3E.103

x5 when x=2

Answer

32

Exercise 1.3E.104

4x when x=2

Exercise 1.3E.105

x2+3x7 when x=4

Answer

21

Exercise 1.3E.106

6x+3y9 when x=10,y=7

Answer

9

Exercise 1.3E.107

(x+y)2 when x=6,y=9

Exercise 1.3E.108

a2+b2 when a=3,b=8

Answer

73

Exercise 1.3E.109

r2s2 when r=12,s=5

Exercise 1.3E.110

2l+2w when l=15,w=12

Answer

54

Exercise 1.3E.111

2l+2w when l=18,w=14

Simplify Expressions by Combining Like Terms

In the following exercises, identify the coefficient of each term.

Exercise 1.3E.112

8a

Answer

8

Exercise 1.3E.113

13m

Exercise 1.3E.114

5r2

Answer

5

Exercise 1.3E.115

6x3

In the following exercises, identify the like terms.

Exercise 1.3E.116

x3,8x,14,8y,5,8x3

Answer

x3 and 8x3, 14 and 5

Exercise 1.3E.117

6z,3w2,1,6z2,4z,w2

Exercise 1.3E.118

9a,a2,16,16b2,4,9b2

Answer

16 and 4, 16b2 and 9b2

Exercise 1.3E.119

3,25r2,10s,10r,4r2,3s

In the following exercises, identify the terms in each expression.

Exercise 1.3E.120

15x2+6x+2

Answer

15x2,6x,2

Exercise 1.3E.121

11x2+8x+5

Exercise 1.3E.122

10y3+y+2

Answer

10y3,y,2

Exercise 1.3E.123

9y3+y+5

In the following exercises, simplify the following expressions by combining like terms.

Exercise 1.3E.124

10x+3x

Answer

13x

Exercise 1.3E.125

15x+4x

Exercise 1.3E.126

4c+2c+c

Answer

7c

Exercise 1.3E.127

6y+4y+y

Exercise 1.3E.128

7u+2+3u+1

Answer

10u+3

Exercise 1.3E.129

8d+6+2d+5

Exercise 1.3E.130

10a+7+5a2+7a4

Answer

22a+1

Exercise 1.3E.131

7c+4+6c3+9c1

Exercise 1.3E.132

3x2+12x+11+14x2+8x+5

Answer

17x2+20x+16

Exercise 1.3E.133

5b2+9b+10+2b2+3b4

Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the phrases into algebraic expressions.

Exercise 1.3E.134

the difference of 14 and 9

Answer

149

Exercise 1.3E.135

the difference of 19 and 8

Exercise 1.3E.136

the product of 9 and 7

Answer

97

Exercise 1.3E.137

the product of 8 and 7

Exercise 1.3E.138

the quotient of 36 and 9

Answer

36\div 9

Exercise \PageIndex{139}

the quotient of 42 and 7

Exercise \PageIndex{140}

the sum of 8x and 3x

Answer

8x+3x

Exercise \PageIndex{141}

the sum of 13x and 3x

Exercise \PageIndex{142}

the quotient of y and 3

Answer

\frac{y}{3}

Exercise \PageIndex{143}

the quotient of y and 8

Exercise \PageIndex{144}

eight times the difference of y and nine

Answer

8(y−9)

Exercise \PageIndex{145}

seven times the difference of y and one

Exercise \PageIndex{146}

Eric has rock and classical CDs in his car. The number of rock CDs is 3 more than the number of classical CDs. Let c represent the number of classical CDs. Write an expression for the number of rock CDs.

Answer

c+3

Exercise \PageIndex{147}

The number of girls in a second-grade class is 4 less than the number of boys. Let b represent the number of boys. Write an expression for the number of girls.

Exercise \PageIndex{148}

Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let n represent the number of nickels. Write an expression for the number of pennies.

Answer

2n - 7

Exercise \PageIndex{149}

Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.

Everyday Math

Exercise \PageIndex{150}

Car insurance Justin’s car insurance has a $750 deductible per incident. This means that he pays $750 and his insurance company will pay all costs beyond $750. If Justin files a claim for $2,100.

  1. how much will he pay?
  2. how much will his insurance company pay?
Answer
  1. $750
  2. $1,350
Exercise \PageIndex{151}

Home insurance Armando’s home insurance has a $2,500 deductible per incident. This means that he pays $2,500 and the insurance company will pay all costs beyond $2,500. If Armando files a claim for $19,400.

  1. how much will he pay?
  2. how much will the insurance company pay?

Writing Exercises

Exercise \PageIndex{152}

Explain the difference between an expression and an equation.

Answer

Answers may vary

Exercise \PageIndex{153}

Why is it important to use the order of operations to simplify an expression?

Exercise \PageIndex{154}

Explain how you identify the like terms in the expression 8a^{2} + 4a + 9 - a^{2} - 1

Answer

Answers may vary

Exercise \PageIndex{155}

Explain the difference between the phrases “4 times the sum of x and y” and “the sum of 4 times x and y.”

Self Check

ⓐ Use this checklist to evaluate your mastery of the objectives of this section.

A table is shown that is composed of four columns and six rows. The header row reads, from left to right, “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The phrases in the first column read “use variables and algebraic symbols.”, “simplify expressions using the order of operations.”, “evaluate an expression.”, “identify and combine like terms.”, and “translate English phrases to algebraic expressions.”

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

More Exponent Practice

Exercise \PageIndex{156}

compute the exact value of the given exponential expression

1. 01

2. 13

3. 32

4. 34

5. 41

6. 52

7. 43

8. 42

9. −32

10. −33

11. −42

12. −24

13. (−5)2

14. (−2)4

15. (−3)3

16. (−2)5

17. (−6)2

18. (−4)2

19. (−5)3

20. (−4)3

21. −(−5)2

22. −(−3)4

23. −(−4)3

24. −(−2)5

Answers to odd problems for E.156

1. 0

3. 9

5. 4

7. 64

 9. −9

11. −64

13. 25

15. −27

17. 36

19. −125

21. −25

23. 64

 

More Order of Operations Practice

Exercise \PageIndex{157}

Compute the exact value of the given expression.

1. 9 − 1(−7)

2. 85 − 8(9)

3. 3 + 9(4)

4. 6 + 7(−1)

5. −2 − 3(−5)

6. 64 − 7(7)

7. 10 + 12(−5)

8. 4 + 12(4)

9. 30 · 5 ÷ 3

10. 72 · 6÷ 4

11. 32 ÷ 4 · 4

12. 64 ÷ 4 · 4

13. 15 ÷ 1 · 3

14. 18 ÷ 6 · 1

15. 40 ÷ 5 · 4

16. 30 ÷ 6 · 5

 

 

19. 10 − 72 ÷ 6 · 3+8

20. 8 − 120 ÷ 5 · 6+7

21. 3 − 24 ÷ 4 · 3+4

22. 4 − 40 ÷ 5 · 4+9

23. 2+6 ÷ 1 · 6 − 1

24. 1 + 12 ÷ 2 · 2 − 6

Answers to odd problems for E.157 

1. 16

3. 39

5. 13

7. −50

9. 50

11. 32

13. 45

15. −216

 

19. −18

21. −11

23. 37

 

More Practice Changing English Expressions into Algebraic Expressions

Exercise \PageIndex{158}

Translate the following phrases into algebraic expressions.

1.    a. the difference of 5x^2 and 6xy

         b. the quotient of 6y^2 and 5x

         c. Twenty-one more than y^2

         d. 6x less than 81x^2

2.    a. the difference of 17x^2 and 17x^2 and 5xy

         b. the quotient of 8y^3 and 3x

         c. Eighteen more than a^2;

         d. 11b less than 100b^2

3.    a. the sum of 4ab^2 and 3a^2b

         b. the product of 4y^2 and 5x

         c. Fifteen more than m

         d. 9x less than 121x^2

4.    a. the sum of 3x^2y and 7xy^2

         b. The product of 6xy^2 and 4z

         c. Twelve more than 3x^2

         d. 7x^2 less than 63x^3

Answers to odd problems for E.158 #1-4

1a. 5x^2−6xy      b. \frac{6y^2}{5x}      c. y^2+21      d. 81x^2−6x          3a. 4ab^2+3a^2b      b. 20xy^2      c. m+15      d. 121x^2−9x.

Translate the following phrases into algebraic expressions.

5.    a. four times the difference of y and six

         b. the difference of four times y and 6

6.    a. five times the difference of y and two 

         b. the difference of five times y and 2

7.    a. five times the sum of 3x and y

         b. the sum of five times 3x and y

8.    a. eleven times the sum of 4x^2 and 5x

         b. the sum of eleven times 4x^2 and 5x

Answers to odd problems for E.158 #5-8

5 a. 4(y−6)      b. 4y−6                  7 a. 5(3x+y)      b. 5(3x)+y.

 

Translate the following phrases into algebraic expressions.

9. The length of a rectangle is 5 inches less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

10. The width of a rectangle is 2 meters greater than the length. Let L  represent the length of the rectangle. Write an expression for the width of the rectangle

11. A collection contains jazz and classical CDs. The number of jazz CDs is 3 more than the number of classical CDs. Let c represent the number of classical CDs. Write an expression for the number of jazz CDs.

12. The number of girls in a second-grade class is 4 less than the number of boys. Let b represent the number of boys. Write an expression for the number of girls.

13. A playlist contains rock and country songs. The number of rock songs is 14 more than twice the number of country songs. Let c represent the number of country songs. Write an expression for the number of rock songs.

14. The number of women in a Statistics class is 8 less than twice the number of men. Let m represent the number of men. Write an expression for the number of women.

15. Greg has nickels and pennies in his pocket. The number of pennies is seven less than three times the number of nickels. Let n represent the number of nickels. Write an expression for the number of pennies.

16. Greg has nickels and pennies in his pocket. The number of pennies is four more than twice the number of nickels. Let n represent the number of nickels. Write an expression for the number of pennies.

Answers to odd problems for E.158 #9-16

9.  w-5     11. c+3     13. 2c+14    15.  3n-7  .

 


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