Exponents Scientific Notation

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Math Skill and Problem-Solving Activities

Skill 9 Defining an Exponent

An exponent shows how many times a base is used as a factor in a product. Exponents may be positive or negative integers or zero.

Example 1 Find the value of 34.

34 5 3 3 3 3 3 3 3

34 5 81

Example 2 Evaluate 1023.

A power with a negative exponent is equal to the reciprocal of that power with a positive exponent.

1023 51

103

1103 5

110 3 10 3 10

110 3 10 3 10 5

11,000 or 0.001

Notice that in the denominator 10 is used as a factor 3 times.

Example 3 270

A power with an exponent of 0 has a value of 1.

270 5 1

TRY iT YouRself!

Write each power as a product of factors. Then evaluate.

1. 104 2. 223 3. 1025

Write each expression as a power. Then evaluate.

4. 4 3 4 3 4 3 4 5. 10 3 10 3 10 6. 110 3 10 3 10 3 10

Find the value of each power.

7. 105 8. 623 9. 1024

10. 50 11. 1022 12. 721

section 4 exponents and scientific Notation

HOW You Will Use this Skill in Science

Understanding the Decibel • Scale Comparing Very Large or Very • Small NumbersUsing the pH Scale•

math vocabulary

A product represented as a base with an exponent is called a power. 43 is a power of 4.

base 43 exponent

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Sound intensity can be measured in watts per square meter (W/m2) or decibels (dB). Decibels are related to watts per square meter as shown in the table. To compare sound levels in decibels, find the difference in the number of decibels. To compare sound levels in watts per square meter, use a ratio.

Sample Problem: Jason wants to know how the sound he experiences riding in a car compares to the sound of talking at the dinner table. Compare the sound intensities using watts per square meter.

1. Read and Understand What is the sound intensity of riding in a car? (1024 W/m2) What is the sound intensity of talking at the dinner table? (1026 W/m2)

2. Plan and SolveWrite a ratio to compare the sound intensities.

1024 W>m2

1026 W>m2

Use reciprocals to rewrite the ratio. 1024

1026 5106

104

Simplify the ratio. 106

104 510 3 10 3 10 3 10 3 10 3 10

10 3 10 3 10 3 10 5 100

The sound intensity for riding in a car is 100 times greater than talking at dinner.

3. Look Back and Check Is the answer reasonable? (Yes, riding in a car is noisier than talking quietly.)

TRY iT YouRself!

1. Find the difference in decibels between using power tools in shop class and relaxing in a quiet room. Then compare the sounds using watts per square meter.

2. Describe any patterns you see in the table of sound intensities.

3. Use the data in the table. Explain why a change of 10 decibels is equal to a tenfold change in watts per square meter.

Skill 9 Problem-Solving Practice: Intensity of Sound

Example Intensity (W/m2) Intensity (dB)

Complete silence 10212 0

Recording studio 10210 20

Relaxing in a quiet room 1028 40

Talking at the dinner table 1026 60

Riding to school in a car 1024 80

Using power tools in shop class

1022 100

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In scientific notation a very large or very small number is written as the product of a decimal and a power of 10. The decimal must be 1.0 or greater but less than 10.

Example 1 Write 6,780,000,000 in scientific notation.

Move the decimal point 9 places to the left to form a number between 1 and 10.

Write the product of 6.78 and a power of 10 with an exponent equal to the number of places the decimal point moved.

Example 2 Write 0.000000501 in scientific notation.

Move the decimal point 7 places to the right to form a number between 1 and 10.

Write the product of 5.01 and a power of 10 with 27 as the exponent.

Example 3 Write 2.076 3 108 in standard form.

Write 2.076. Move the decimal point 8 places to the right since the exponent is 8. Add commas.

Example 4 Write 6.108 3 1026 in standard form.

Write 6.108. Move the decimal point 6 places to the left since the exponent is 26. Place a 0 before the decimal point.

TRY iT YouRself!

Write each number in scientific notation.

1. 56,782,500,000 2. 0.00000807 3. 100.001

Write each number in standard form.

4. 2.308 3 108 5. 4.009 3 1025 6. 8.0901 3 105

Skill 10 Expressing Numbers in Scientific Notation


Describing Very Large • Distances as Used in Astronomy Describing Very Small Distances • as Used in Molecular Research

6. 7 8 0 0 0 0 0 0 0. S 6.78

6.78 3 109

0 . 0 0 0 0 0 0 5. 0 1 S 5.01

5.01 3 1027

2 . 0 7 6 0 0 0 0 0 . S 207,600,000

. 0 0 0 0 0 6 . 1 0 8 S 0.000006108

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Scientists work with different units of distance depending on the nature of their work. Some examples of units of distance are listed in the table at the right.

Task Unit DistanceFinding distances in the solar system astronomical unit (AU) 149,597,870 km

Measuring distances between atoms angstrom (Å) 10210 m

Finding nautical depths fathom 6 ft

Measuring distances in galaxies light-year 9.4607 3 1012 km

Sample Problem: Ana rounded an astronomical unit to the nearest million kilometers and wrote it in scientific notation as 15 3 107 km. Zena said it is 1.5 3 108 km. Who is right? Why?

1. Read and Understand What is an astronomical unit? (149,597,870 km) What is scientific notation? (a way to write numbers using powers of 10)

2. Plan and Solve Round 149,597,870 km to the nearest million kilometers. 150,000,000 km

Write 150,000,000 in scientific notation. 1.5 3 108

Find the person who answered correctly. Zena

Explain why the other person is wrong. The decimal factor in scientific notation must be between 1 and 10.

3. Look Back and Check Does the answer make sense? (Yes, the answer uses the correct format for scientific notation.)

TRY iT YouRself!

1. Which distance in the table is written in scientific notation? Explain.

2. Write the distance of a fathom in scientific notation.

3. Use standard form and scientific notation to write the number of meters in an angstrom.

4. Write the number of kilometers in a light-year in standard form.

5. Which unit represents the greatest distance? Explain your reasoning.

Skill 10 Problem-Solving Practice: Units of Distance

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To add or subtract numbers in scientific notation, make sure that the exponents in the powers of 10 are the same. Add or subtract the decimal factors. Write the sum or difference as a product with the power of 10. Check that the answer is still in scientific notation. Rewrite if needed.

Example 1 (3.230 3 105) 1 (6.034 3 105)

The exponents are the same. Add 3.23 and 6.034.

Write the product of 9.264 and 105.

Example 2 (7.23 3 108) 2 (6.89 3 108)

The exponents are the same. Subtract 6.89 from 7.23.


Rewrite in scientific notation.

Example 3 (1.03 3 104) 1 (2.089 3 105)

The exponents are not the same. Rewrite, using the same exponents.

Add 0.103 and 2.089.


TRY iT YouRself!

Add or subtract.

1. (7.6 3 105) 1 (3.2 3 105) 2. (9.09 3 1024) 2 (2.9 3 1024)

3. (5.4 3 109) 1 (6.02 3 109) 4. (2.308 3 108) 2 (3.33 3 107)

5. (4.08 3 1025) 2 (5.9 3 1026) 6. (8.0901 3 103) 1 (2.22 3 102)

Skill 11 Addition and Subtraction in Scientific Notation

3.2301 6.034 9.264

9.264 3 105

7.232 6.89 0.34

0.34 3 108

3.4 3 107

1.03 3 104 5 0.103 3 105

0.1031 2.089 2.192

2.192 3 105


Finding Distances between • Planets in the Solar System Describing the Sizes of Atoms • and Molecules

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The average distances from the Sun and the masses of the planets in the solar system are large numbers that are best expressed in scientific notation.

Sample Problem: Use the table. Find the combined mass of the three planets closest to the Sun. Write the answer.

1. Read and Understand What are the three planets closest to the Sun? (Mercury, Venus, Earth)

What operation is used to find the combined mass? (addition)

2. Plan and Solve Rewrite so the exponents are the same. Then add.

Check whether the answer is written in scientific notation. Rewrite if needed.

Label the answer with the correct unit.

3. Look Back and Check Does the answer make sense? (Yes, rounding the masses and adding gives 1.1 3 1025 kg which is close to 1.1176 3 1025 kg.)

TRY iT YouRself!

Write each answer in scientific notation.

1. What is the difference in mass between the planet with the greatest mass and the planet with the least mass?

2. What is the average distance between Earth and Mars?

3. What is the average distance between Venus and Uranus?

4. Use the data in the table to write two problems that involve addition and subtraction with scientific notation.

Skill 11 Problem-Solving Practice: Planets of the Solar System

Planet Average Distance

from Sun (km)Mass (kg)

Mercury 5.8 3 107 3.30 3 1023

Venus 1.08 3 108 4.87 3 1024

Earth 1.50 3 108 5.976 3 1024

Mars 2.28 3 108 6.418 3 1023

Jupiter 7.78 3 108 1.899 3 1027

Saturn 1.430 3 109 5.685 3 1026

Uranus 2.870 3 109 8.684 3 1025

Neptune 4.500 3 109 1.02 3 1026

3.30 3 1023 5 0.330 3 1024

4.87 3 1024 5 4.87 3 1024

1 5.976 3 1024 5 5.976 3 1024

11.176 3 1024

11.176 3 1024 5 1.1176 3 1025

1.1176 3 1025 kg

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To multiply in scientific notation, first multiply the decimal factors. Then multiply the powers of 10 by adding the exponents. Write the new product in scientific notation.

Example 1 (4.3 3 105) 3 (5.04 3 103)

Multiply the decimal factors.

Multiply the powers by adding the exponents.

Write the product in scientific notation.

To divide in scientific notation, first divide the decimal factors. Then divide the powers of 10 by subtracting the exponents. Write the new product in scientific notation.

Example 2 (7.20 3 108) 4 (1.89 3 104)

Divide the decimal factors. Round the quotient to the nearest hundredth.

Divide the powers by subtracting the exponents.

Write the quotient in scientific notation.

TRY iT YouRself!

Multiply or divide.

1. (3.7 3 105) 3 (6.3 3 104) 2. (1.595 3 104) 4 (2.9 3 105)

3. (3.2 3 107) 3 (5.01 3 105) 4. (3.09 3 106) 4 (1.03 3 1028)

5. (7.777 3 103) 4 (1.1 3 1027) 6. (9.01 3 105) 3 (2.22 3 1023)

Skill 12 Multiplication and Division in Scientific Notation

4.3 3 5.04 5 21.672

105 3 103 5 10513 5 108

21.672 3 108 5 2.1672 3 109

7.20 4 1.89 < 3.81

108 4 104 5 10824 5 104

3.81 3 104


Finding Relationships Among • Distance Units Using the Wave Equation•

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Radio waves are electromagnetic waves that are used for many kinds of communication, including radio and television broadcasts. The speed, wavelength, and frequency of a radio wave are related by this equation:

Speed 5 Wavelength 3 Frequency

In space, radio waves travel at the speed of light, or about 3 3 108 m/s.

Sample Problem: An amateur radio operator broadcasts at a frequency of

1.818 3 106 Hz. What is the wavelength of the broadcast?

1. Read and Understand What equation is needed to solve this problem? (Wavelength 5

SpeedFrequency)

What frequency is the amateur radio operator using? (1.818 3 106 Hz)

2. Plan and Solve

Write the equation.

Substitute the known values. Be sure to include units.

Solve the equation. (Hint: 1 Hz 5 1s 5 s21)

Label the answer with the correct unit.

3. Look Back and Check

Is the computation done correctly? Is the unit in the answer correct?

(Yes, 3 3 108

1.818 3 106 is close to 3 3 108

2 3 106, or 150. Unit analysis shows no errors.)

TRY iT YouRself!

1. A local television station broadcasts at a wavelength of 6 m. At what frequency is it broadcasting?

2. A satellite communication system transmits at 6 3 109 Hz. What is the

wavelength of the transmission?

3. An AM radio station broadcasts at 1.2 3 106 Hz. At what wavelength does this

station broadcast?

Skill 12 Problem-Solving Practice: Exploring Radio Waves

Wavelength 5Speed

Frequency

Wavelength 53 3 108 m>s

1.818 3 106 Hz

Wavelength 53 3 108 m>s

1.818 3 106 s21

5 3 3 108

1.818 3 106 m

5 31.818 3 10826 m

< 1.65 3 102 m

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When do elephants have eight feet?

To find out, locate the correct answers at the bottom of the page and shade the letters above them. The remaining letters answer the riddle. (Hint: Some answers may be written as decimals.)

1. Evaluate: 922 8. Evaluate: 34

2. (1.25 3 102) 3 (8.0 3 103) 5 9. (7.5 3 106) 4 (2.5 3 102) 5

3. Evaluate: 103 5 10. (1.5 3 106) 4 (3.0 3 1023) 5

4. (2.5 3 106) 3 (2.0 3 103) 5 11. (3.5 3 1023) 1 (2.5 3 1023) 5

5. (1.25 3 105) 3 (4.0 3 102) 5 12. (3.5 3 105) 3 (2.0 3 1023) 5

6. Evaluate: 1025 5 13. Evaluate: 323

7. (9.5 3 108) 2 (2.5 3 108) 5 14. Evaluate: 1023 5

W A H C E N T O T H

210

11,000 6 3 103 1

81 5 3 105 19 6 3 1023 1 3 106 27 100

P E S R E D L A U R

5 3 108 4 3 103 1,000 7 3 109 1 127 81 1

2 0.00001 8

E N M T H W R I O !

50 7 3 102 5 3 109 7.0 5 3 107 100,000 7 3 108 3 3 104 48

Section 4 Review

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Solve each problem.

1. Complete silence has a sound intensity

of 10212 W/m2. Riding in a car has a sound intensity of 1024 W/m2. How many times greater is the sound intensity in a car than complete silence?

2. There are approximately 6.02 3 1023 atoms of carbon-12 in 12 grams of carbon-12. About how many atoms of carbon-12 would there be in 60 grams?

3. The mass of Jupiter is 1.899 3 1027 kg. The mass of Saturn is 5.685 3 1026 kg. Which planet has the greater mass? How much greater?

4. The diameter of a water molecule is about 0.00000000028 m. Write the diameter using scientific notation.

5. Venus has a mass of 4.87 3 1024 kg and

Earth has a mass of 5.976 3 1024 kg. What is the combined mass of the two planets?

6. What is the frequency of an electromagnetic wave with a wavelength of 5 meters? Remember, electromagnetic waves travel at a speed

of about 3 3 108 m/s.

Use the Table for Problems 7–9.

Unit Distance (km)

Astronomical unit 149.6 3 106 km

Light-year 9.46 3 1012 km

Parsec 3.09 3 1013 km

Hint: Round decimals to the nearest thousandth in your answers.

7. Find the number of astronomical units in a light-year.

8. Find the number of astronomical units in a parsec.

9. Find the number of light-years in a parsec.

Problem-Solving Practice for Section 4

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Exponents Scientific Notation