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: $2⋅43$

8. $78$

9. $455$

Answer: $5⋅7⋅13$

10. $400$

11. $432$

Answer: $2⋅2⋅2⋅2⋅3⋅3⋅3$

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. $2^3−12÷(9−5)$

Answer: $5$

20. $3^2−18÷(11−5)$

21. $2+8(6+1)$

Answer: $58$

22. $4+6(3+6)$

23. $20÷4+6(5−1)$

Answer: $29$

24. $33÷3+4(7−2)$

25. $3(1+9⋅6)−4^2$

Answer: $149$

26. $5(2+8⋅4)−7^2$

27. $2[1+3(10−2)]$

Answer: $50$

28. $5[2+4(3−2)]$

29. $8+2[7−2(5−3)]−3^2$

Answer: $5$

30. $10+3[6−2(4−2)]−2^4$

Evaluate an Expression

In the following exercises, evaluate the following expressions.

31. When $x=2$,

a. $x^6$

b. $4^x$

c. $2x^2+3x−7$

Answer: a. 64
b. 16
c. 7

32. When $x=3$,

a. $x^5$

b. $5x$

c. $3x^2−4x−8$

33. When $x=4$ and $y=1$

$x^2+3xy−7y^2$

Answer: $21$

34. When $x=3$ and $y=2$

$6x^2+3xy−9y^2$

35. When $x=10$ and $y=7$

$(x−y)^2$

Answer: $9$

36. When $a=3$ and $b=8$

$a^2+b^2$

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. $8y+5+2y−4$

39. $10a+7+5a−2+7a−4$

Answer: $22a+1$

40. $7c+4+6c−3+9c−1$

41. $3x^2+12x+11+14x^2+8x+5$

Answer: $17x^2+20x+16$

42. $5b^2+9b+10+2b^2+3b−4$

Translate an English Phrase to an Algebraic Expression

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

43. 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 $81x^2$

Answer

a. $5x^2−6xy$ b. $\frac{6y^2}{5x}$

c. $y^2+21$ d. $81x^2−6x$

44. 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 $100b^2$

45. 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 $121x^2$

Answer

a. $4ab^2+3a^2b$ b. $20xy^2$

c. $m+15$ d. $121x^2−9x$

46. 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 $63x^3$

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

b. the difference of eight times $y$ and $9$

Answer: a. $8(y−9)$
b. $8y−9$

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 $4x^2$ and $5x$

b. the sum of eleven times $4x^2$ 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: $2c-14$

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: $3n-7$

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 $8a^2+4a+9−a^2−1.$

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.

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Practice Makes Perfect

Writing Exercises

Self Check