1B Formulas with Multiplying Whole Numbers and Positive Exponents

A common formula in geometry classes is the volume of a box (rectangular prism). The volume formula is $V=L\times W\times H$. If we knew the length, width and height of the box, could we calculate the volume? Again, many students miss this problem due to poor arithmetic skills. So let’s review multiplication and positive exponents so we can make our Geometry teachers happy.

Remember multiplication is repeated addition. If I have 17 buses each with 45 students in each of them, then I have 45 + 45 +45 + … (17 times). An easy way to figure this out is to multiply 45 x 17 = 765 total students.

Let’s start by reviewing multiplication by one and zero. Recall that any number times 1 is itself. So 53 x 1 = 53 and 1 x 728 = 728. Also recall that any number times zero is zero. So 37 x 0 = 0 and 0 x 74 = 0.

Next let’s review multiplying numbers with lots of zeros. There is an easy shortcut to this. For nonzero whole-number factors, remove only trailing zeros, multiply the remaining factors, then append their combined number of trailing zeros. For example let’s look at 12000 x 400. Start with 12 x 4 = 48. We have an additional five zeros so the answer is 4,800,000.

Multiplication can be on the complicated side due to partial products. Let’s look at 36 x 52 for example. Many students think that multiplication works like addition and we simply multiply the place values. That is an extremely wrong idea. 36 x 52 is really (30+6)(50+2) = (30x50)+(30x2) + (6x50) + (6x2). These are called partial products. So the answer is 1500 + 60 + 300 + 12 = 1872. Common algorithms for multiplication can also multiply (2x36) + (50x36) and then add.

Do the following examples with your instructor. Pay close attention to the partial products.

Many formulas involve exponents. A positive whole-number exponent gives the number of copies of the base used as factors. For example, an exponent of 1 means one copy of the base, with no multiplication operation needed. $7^4$ means 7 x 7 x 7 x 7 (multiply four 7’s). So $7^4=2401$.

Do the following examples with your instructor. Pay close attention. Which number is the base and which is the exponent?

Practice Problems Section 1B

Perform the indicated operation for #1-54.

1. 4 x 13

   Check answer

   52

2. 52 x 8

3. 14 x 19

   Check answer

   266

4. 300 x 60

5. 240 x 1,000

   Check answer

   240,000

6. 5,000 x 1,400

7. 24 x 79

   Check answer

   1,896

8. 51 x 64

9. 31 x 569

   Check answer

   17,639

10. 120 x 45,000

11. 135 x 892

    Check answer

    120,420

12. 206 x 784

13. 26 x 4,018

    Check answer

    104,468

14. 164 x 37

15. 71 x 928

    Check answer

    65,888

16. 30,000 x 19,000

17. 400 x 132,000

    Check answer

    52,800,000

18. 734 x 906

19. 7 x 12 x 18

    Check answer

    1,512

20. 56 x 4 x 10

21. 29 x 135

    Check answer

    3,915

22. 1,700 x 50

23. 3600 x 1,000

    Check answer

    3,600,000

24. 72,000 x 300

25. 301 x 506

    Check answer

    152,306

26. 624 x 207

27. 8 x 2005

    Check answer

    16,040

28. 120 x 0

29. 1 x 934

    Check answer

    934

30. 267 x 0

31. 574 x 1

    Check answer

    574

32. 0 x 75

33. 1 x 9,734

    Check answer

    9,734

34. 40,000 x 13,000

35. 600 x 108,000

    Check answer

    64,800,000

36. 31,000 x 4,000

37. $$9^2$$

    Check answer

    81

38. $$5^3$$

39. $$6^2$$

    Check answer

    36

40. $$6^3$$

41. $$2^5$$

    Check answer

    32

42. $$3^4$$

43. $${13}^2$$

    Check answer

    169

44. $${11}^3$$

45. $${25}^2$$

    Check answer

    625

46. $$1^6$$

47. $$0^4$$

    Check answer

    0

48. $$8^2\times 3$$

49. $$7\times 2^4$$

    Check answer

    112

50. $$7^2\times 3$$

51. $$3^2\times 5\times 4$$

    Check answer

    180

52. $$11\times 3^3$$

53. $$5^3\times 17$$

    Check answer

    2,125

54. $${16}^2\times 10\times 8$$

55. The perimeter of a square is given by the formula $P=4\times s$, where s is the length of one of the sides of the square. What is the perimeter of a square window where the length of a side is 24 inches?

    Check answer

    96 in

56. The perimeter of a square is given by the formula $P=4\times s$, where s is the length of one of the sides of the square. What is the perimeter of a square external hard-drive for a computer where the length of a side is 27 mm?

57. The perimeter of a square is given by the formula $P=4\times s$, where s is the length of one of the sides of the square. What is the perimeter of a square-shaped garden where the length of a side is 113 feet?

    Check answer

    452 ft

58. Use the volume of a box formula $V=L\times W\times H$ to calculate the volume of a box that is 4 feet by 6 feet by 13 feet.

59. Use the volume of a box formula $V=L\times W\times H$ to calculate the volume of a box that is 12 in by 8 in by 15 in.

    Check answer

    1,440 cubic in

60. Use the volume of a box formula $V=L\times W\times H$ to calculate the volume of a box that is 23 cm by 16 cm by 100 cm.

61. The volume of a cube is given by the formula $V=s^3$ where s is the length of a side of the cube. Calculate the volume of a cube that has a side that is 8 mm long.

    Check answer

    512 cubic mm

62. The volume of a cube is given by the formula $V=s^3$ where s is the length of a side of the cube. Calculate the volume of a cube that has a side that is 20 inches long.

63. Leah started a savings account. Every month $145 will be deposited in her account. How much total money will she have deposited after 6 months?

    Check answer

    $870

64. Brian rents a room for $575 per month. If he paid the rent for 8 months, how much total rent would he have paid?

65. The standard deviation in a data set is 26 pounds. How much is three standard deviations?

    Check answer

    78 pounds

66. The standard deviation in a data set is 17 inches. How far is four standard deviations?

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.