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1.1E: Exercises

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Practice Makes Perfect

Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.

1. 84

Answer

Divisible by 2, 3, 6

2. 96

3. 896

Answer

Divisible by 2

4. 942

5. 22,335

Answer

Divisible by 3, 5

6. 39,075

Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

7. 86

Answer

243

8. 78

9. 455

Answer

5713

10. 400

11. 432

Answer

2222333

12. 627

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

13. 8,12

Answer

24

14. 12,16

15. 28,40

Answer

280

16. 84,90

17. 55,88

Answer

440

18. 60,72

Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

19. 2312÷(95)

Answer

5

20. 3218÷(115)

21. 2+8(6+1)

Answer

58

22. 4+6(3+6)

23. 20÷4+6(51)

Answer

29

24. 33÷3+4(72)

25. 3(1+96)42

Answer

149

26. 5(2+84)72

27. 2[1+3(102)]

Answer

50

28. 5[2+4(32)]

29. 8+2[72(53)]32

Answer

5

30. 10+3[62(42)]24

Evaluate an Expression

In the following exercises, evaluate the following expressions.

31. When x=2,

a. x6

b. 4x

c. 2x2+3x7

Answer

a. 64
b. 16
c. 7

32. When x=3,

a. x5

b. 5x

c. 3x24x8

33. When x=4 and y=1

x2+3xy7y2

Answer

21

34. When x=3 and y=2

6x2+3xy9y2

35. When x=10 and y=7

(xy)2

Answer

9

36. When a=3 and b=8

a2+b2

Simplify Expressions by Combining Like Terms

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

37. 7x+2+3x+4

Answer

10x+6

38. 8y+5+2y4

39. 10a+7+5a2+7a4

Answer

22a+1

40. 7c+4+6c3+9c1

41. 3x2+12x+11+14x2+8x+5

Answer

17x2+20x+16

42. 5b2+9b+10+2b2+3b4

Translate an English Phrase to an Algebraic Expression

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

43. a. the difference of 5x2 and 6xy

b. the quotient of 6y2 and 5x

c. Twenty-one more than y2

d. 6x less than 81x2

Answer

a. 5x26xy b. 6y25x

c. y2+21 d. 81x26x

44. a. the difference of 17x2 and 17x2 and 5xy

b. the quotient of 8y3 and 3x

c. Eighteen more than a2;

d. 11b less than 100b2

45. a. the sum of 4ab2 and 3a2b

b. the product of 4y2 and 5x

c. Fifteen more than m

d. 9x less than 121x2

Answer

a. 4ab2+3a2b b. 20xy2

c. m+15 d. 121x29x 

46. a. the sum of 3x2y and 7xy2

b. The product of 6xy2 and 4z

c. Twelve more than 3x2

d. 7x2 less than 63x3

47. a. eight times the difference of y and nine

b. the difference of eight times y and 9

Answer

a. 8(y9)
b. 8y9

48. a. seven times the difference of y and one

b. the difference of seven times y and 1

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

b. the sum of five times 3x and y

Answer

a. 5(3x+y)
b. 15x+y

50. a. eleven times the sum of 4x2 and 5x

b. the sum of eleven times 4x2 and 5x

51. Eric has rock and country songs on his playlist. 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.

Answer

2c14

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

53. 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.

Answer

3n7

54. 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.

Writing Exercises

55. Explain in your own words how to find the prime factorization of a composite number.

Answer

Answers will vary.

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

57. Explain how you identify the like terms in the expression 8a2+4a+9a21.

Answer

Answers will vary.

58. 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

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

This table has 4 columns, 7 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: identify multiples and apply divisibility tests, find prime factorizations and least common multiples, use variables and algebraic symbols, simplify expressions using the order of operations, evaluate an expression, identify and combine like terms, translate English phrases to algebraic expressions. The remaining columns are blank.

b. If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.


This page titled 1.1E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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