Power of 10

A number that indicates to what power the factor is being raised.

The process of performing mathematical calculations in one’s head without paper and pencil.

A prescribed decimal place, which determines the amount of rounding off to be done; this is usually based on the degree of accuracy needed in measurement.

Standard Scientific Notation

A method of writing or displaying numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10.

Objective 1

Many students find larger numbers or numbers with decimals intimidating. As we have discussed, the larger the number, the greater the chance of error—and errors can affect a patient’s rate of recovery. In this chapter, we discuss strategies for reducing large numbers by using the principles of power of 10 or writing numbers in scientific notation. Exponents are used to represent large base 10 numbers and when writing numbers in scientific notation. If you are unsure of how to manipulate exponents or negative numbers, you might want to review the relevant sections in Chapter 1.

In Chapter 7, you learned the prefixes that are used in the metric system. Because the metric system is based on the power of 10, it is common to solve the conversion problems using mental math or manipulation of the decimal point. Table 10-1 demonstrates the power of 10 written as an exponent and as the multiplier.

Table 10-1

Metrix Prefixes as Power of 10

Prefix	Multiplier	Power of 10
Micro-	0.000001	10^-6
Milli-	0.001	10^-3
Centi-	0.01	10^-2
Deci-	0.1	10^-1
Deka-	10	10
Hecto-	100	10²
Kilo-	1,000	10³

Because the metric system is based on the factor 10, conversion problems are often solved by simply moving the decimal point to either the right or the left.

PRACTICE THE SKILL 10-1

Write the following numbers as exponents

1. 100 _________________________________________________

2. 10,000 _________________________________________________

3. 1,000,000 _________________________________________________

4. 1,000 _________________________________________________

5. 100,000 _________________________________________________

6. 0.001 _________________________________________________

7. 0.000001 _________________________________________________

8. 0.00001 _________________________________________________

9. 0.01 _________________________________________________

10. 0.000000001___________________________________________________

Objective 3

Multiplication and division by the power of 10

Multiplying and dividing by the power of 10 can be confusing because the process involves addition or subtraction of the exponents. When you are working with the same base number, like 10, the answer can be obtained by adding the exponents. Division is the reverse; to obtain the answer, subtract the exponents.

MATH ETIQUETTE 10-1

The Law of Exponents states that when you are multiplying or dividing two or more powers with the same base, the answer can be obtained by adding (for multiplication) or subtracting (for division) the exponents.

Example:
100 × 1,000 1. Write the equation as a power of 10. 10² × 10³ 2. The operation is multiplication; add the exponents to obtain the answer. 10²⁺³ 3. The answer is 10⁵.
Example:
10,000 ÷ 100 1. Write the equation as a power of 10. 10⁴/10² 2. The operation is division; subtract the exponents to obtain the answer. 10^4-2 3. The answer is 10².

PRACTICE THE SKILL 10-2

Rewrite the problems using the power of 10. Solve the problem, keeping your answer in the power of 10 format

1. 10 × 100 __________________________

2. 1,000 × 10,000 __________________________

3. 100 × 100,000 __________________________

4. 10 × 10 × 100 __________________________

5. 1,000 × 1,000,000 __________________________

6. 1,000 × 100 × 10 __________________________

7. 0.1 × 0.001 __________________________

8. 0.00001 × 10 __________________________

9. 0.001 × 0.000001 __________________________

10. 0.001 × 0.0001 × 0.1 __________________________

11. 1,000,000 ÷ 1,000 __________________________

12. 100 ÷ 1,000,000 __________________________

13. 10 ÷ 0.001 __________________________

14. 0.01 ÷ 0.001 __________________________

15. 100 ÷ 10,000 __________________________

16. 0.001 ÷ 100 __________________________

17. 0.00000001 ÷ 0.00001 __________________________

18. 100,000 ÷ 100,000,000 __________________________

19. 100,000,000 ÷ 1,000 __________________________

20. 0.001 ÷ 100 __________________________

Another way to calculate with the power of 10 is to count the number of 0s and move your decimal point the same amount to the right (multiply) or left (division). Like many people, you may be able to perform this operation in your head. However, explaining how you solved the problem can be difficult. Let’s break down the steps that you may be doing in your head.

STRATEGY 10-1
Multiplication of Powers

1. Identify which operation is being performed—multiplication or division.

2. Count the number of 0s in the multiplier.

3. Move the decimal point that number of places to the right.

Example:
14.118 × 100
1. Identify which operation is being performed—multiplication or division. multiplication 2. Count the number of 0s in the multiplier. 100 is the multiplier, and it has two 0s. 3. Move the decimal point two places to the right. 14.118 = 1411.8 The correct answer is 1411.8.

STRATEGY 10-2
Division of Powers

1. Identify which operation is being performed—multiplication or division.

2. Count the number of 0s in the divisor.

3. Move the decimal point that number of places to the left.

Example:
310199.1 ÷ 10,000
1. Identify which operation is being performed—multiplication or division. division 2. Count the number of 0s in the divisor. 10,000 is the divisor, and there are four 0s. 3. Move the decimal point four places to the left. 310199.1
The correct answer is 31.01991.

PRACTICE THE SKILL 10-3

Complete the following table

Do not use a calculator; this is a mental math skill.

Problem	Which Direction Is the Decimal Moving?	How Many Places Will the Decimal Move?	Answer
1. 892.0491 × 100
2. 45.00872 × 1000
3. 275632.1 ÷ 100,000
4. 517199.5 ÷ 1000
5. 11.44 × 10
6. 92519.64 ÷ 100
7. 621.3589 × 100
8. 6213.589 ÷ 100
9. 321.557 × 10
10. 873259.3 ÷ 100,000

Conversion Between Metric Units

Chapter 5 discussed the way to set up proportions and ratios so that you could convert between metric measurements to find the equivalent value. Since the metric system is based on the power of 10, many students find it easier to convert similar measurements by manipulating the decimal point rather than setting up the problem as a proportion. When the conversion is from a larger to a smaller metric unit, the decimal point moves to the right. When a smaller metric unit is being converted to a larger one, the decimal point moves to the left (Table 10-2).

Table 10-2

Movement of Decimal Points

Direction of Movement in Problem	Movement of Decimal	Multiply by Multiplier?	Divide by Multiplier?
From a larger metric unit to a smaller metric unit	Right	Yes	No
From a smaller metric unit to a larger metric unit	Left	No	Yes

MATH IN THE REAL WORLD 10-1
Conversions Found in Health Care

• Millimeters to centimeters

• Centimeters to meters

• Milliliters to liters

• Milligrams to grams

• Grams to kilograms

See Figure 7-1, p. 168.

Example:
How many micrograms are in 0.5 mg>? 1. We know that micrograms are smaller than milligrams, so move the decimal to the right. 0.500 2. Micrograms are represented as 10^-6. 3. Milligrams are represented as 10^-3. 4. Move the decimal over three places to the right. 5. Answer: There are 500 mcg in 0.5 mg.

Many times people have difficulty not with the actual calculation, but with the proper way to set up the problem. In the next exercise, the goal is to assist you in applying strategies that will build your confidence in setting up math problems.

BUILDING CONFIDENCE WITH THE SKILL 10-1

Complete the following table

Problem	Movement of Decimal: Right or Left	Mathematical Operation: Multiplication or Division
1. 0.15 g to mg
2. 0.25 ml to dl
3. 500 mg to g
4. 1.75 cm to mm
5. 5 mm to m
6. 2 L to dl
7. 3500 ml to L
8. 0.04 mg to mcg
9. 2.5 g to mg
10. 1500 ml to L
11. 300 hm to m
12. 4.5 km to m
13. 750 mg to g
14. 800 mcg to g
15. 2 L to ml

Objective 4

Significant figures

To what place value should answers be rounded? I have often been asked this question during my teaching career. My usual response is to round to the place value that makes sense and answers the question.

Example:
While at the grocery store, Ashley purchases the following items:
Bananas	3 pounds	Cost: $0.59/pound
Bread	2 loaves	$1.89/loaf
Potatoes	5 pound bag	$1.50/pound
Chicken soup	3 cans	$0.79/can
What is the total cost of the bill before sales tax? What is the price of the bill with a 6.5% sales tax?