2B Formulas with Exponents and Multiplying Decimals

Look at the following formula. $A=\pi r^2$ This is a famous formula in geometry that gives the area of a circle with radius r. Remember that $\pi \approx 3.14$. Suppose we want to find the area of a circle with a radius of 1.5 cm. We will need to review multiplying decimals as well as be able to use exponents with decimals.

Let’s start with a review of multiplying decimals. We start by multiplying the numbers as if they were whole numbers. We then count the total number of decimal places to the right of the decimal in both of the numbers being multiplied. The answer will need the same total number of decimals places to the right of the decimal. For example let’s look at 3.5 x 0.81. First we need to multiply 35 x 81. Remember to use partial products. 35 x 81 = 2835. The question now is where to place the decimal? 3.5 has one place to the right of the decimal and 0.81 has two. So we have a total of 3 decimals to the right of the decimal. Hence our answer should also have 3 decimal places. So 2.835 is the answer. Notice that unlike addition and subtraction, we do not need to line up the place values when we multiply.

Occasionally we may want to multiply a decimal by a power of 10 (10, 100, 1000, 10000, etc.) When we multiply a decimal by one of these numbers the decimal point moves to the right the same number of places as the number of zeros in the numbers. For example 3.4 x 1000, the decimal point will move 3 places to the right and we get 3400. Another example is 0.0000754 x 100000. Notice the decimal point will move 5 places to the right and we get 7.54 as our answer.

Try the following examples with your instructor.

For the positive whole-number exponents used here, exponents mean repeated multiplication. So ${1.3}^2$ means 1.3 x 1.3 = 1.69. Here is another example: ${0.2}^3$ = 0.2 x 0.2 x 0.2 = 0.008 (answer).

Try the following examples with your instructor.

Practice Problems Section 2B

For #1-57, simplify any exponents and then multiply.

1. 0.038 x 100

   Check answer

   3.8

2. 1.42 x 1000

3. 218.3 x 10

   Check answer

   2183

4. 0.00073 x 10000

5. 4.56 x 100

   Check answer

   456

6. 0.00226 x 1000

7. 1.29 x 1000

   Check answer

   1290

8. 0.000352 x 1000

9. 0.0358 x 100

   Check answer

   3.58

10. 10 x 0.348

11. 10000 x 0.00057

    Check answer

    5.7

12. 100 x 34.711

13. 1000 x 0.336

    Check answer

    336

14. 100 x 0.9174

15. 0.3791 x 10

    Check answer

    3.791

16. ${10}^6$ (this is called a million and the prefix “mega” is used)

17. ${10}^9$ (this is called a billion and the prefix “giga” is used)

    Check answer

    1,000,000,000

18. ${10}^{12}$ (this is called a trillion and the prefix “tera” is used)

19. $$45.38\times {10}^8$$

    Check answer

    4,538,000,000

20. $$7.153\times {10}^{11}$$

21. $$721.68\times {10}^6$$

    Check answer

    721,680,000

22. $$71.88\times {10}^7$$

23. $$0.0486\times {10}^{12}$$

    Check answer

    48,600,000,000

24. $$0.9314\times {10}^9$$

25. $$884.62\times {10}^5$$

    Check answer

    88,462,000

26. $$0.005482\times {10}^{10}$$

27. $$309.6\times {10}^4$$

    Check answer

    3,096,000

28. 0.007 x 0.9

29. 0.012 x 0.07

    Check answer

    0.00084

30. 0.016 x 0.09

31. 8 x 1.75

    Check answer

    14

32. 1.3 x 4.6

33. 0.007 x 0.24

    Check answer

    0.00168

34. 0.136 x 8.2

35. 0.006 x 0.017

    Check answer

    0.000102

36. 19.4 x 13.7

37. 7.4 x 0.31

    Check answer

    2.294

38. 0.27 x 3.9

39. 41.8 x 0.006

    Check answer

    0.2508

40. 0.096 x 1.33

41. 12.7 x 0.044

    Check answer

    0.5588

42. 212.3 x 0.0082

43. 0.6 x 0.3 x 1.2

    Check answer

    0.216

44. 0.043 x 10 x 0.7

45. 0.14 x 8 x 0.03

    Check answer

    0.0336

46. $${0.08}^2$$

47. $${0.2}^4$$

    Check answer

    0.0016

48. $${0.01}^3$$

49. $${1.9}^2$$

    Check answer

    3.61

50. $${0.5}^3$$

51. $${0.09}^2$$

    Check answer

    0.0081

52. $$3.5\times {0.3}^3$$

53. $$0.007\times {4.1}^2$$

    Check answer

    0.11767

54. $${0.6}^2\times {0.4}^2$$

55. $${0.8}^3\times 0.14$$

    Check answer

    0.07168

56. $$0.03\times {0.5}^2$$

57. $${0.1}^2\times {0.4}^3$$

    Check answer

    0.00064

Use the formula$A=\pi r^2$to find the area of the following circles. (Remember $\pi \approx 3.14$ and we should always do exponents before multiplying.)

58. Find the area of a circle with a radius of 3.2 feet.

59. Find the area of a circle with a radius of 1.4 mm.

    Check answer

    6.1544 sq mm

60. Find the area of a circle with a radius of 7.5 inches.

61. Find the area of a circle with a radius of 2.9 cm.

    Check answer

    26.4074 sq cm

A company makes a monthly profit of $2800.40. The total profit after m months is given by the formula T = 2800.4 x m .

62. Find the total profit after 12 months (1 year).

63. Find the total profit after 36 months (3 years).

    Check answer

    $100,814.40

Find the volume of a cube with the formula $V=s^3$ where V is the volume and s is the length of a side of the cube.

64. Find the volume of a cube if one side of the cube is 0.7 feet long.

65. Find the volume of a cube if one side of the cube is 1.2 cm long.

    Check answer

    1.728 cu cm

The volume of a can (circular cylinder) is given by the formula $V=\pi r^{2}h$ where r is the radius and h is the height. (Use $\pi \approx 3.14$)

66. Find out how much water a water tank can hold if the radius of the tank is 1.5 meters and the height is 10 meters. (Use $\pi \approx 3.14$)

67. What is the volume of a cylinder shaped trash can that has a radius of 2.1 feet and a height of 4 feet? (Use $\pi \approx 3.14$)

    Check answer

    55.3896 cu ft

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