Download Scientific Notation

Scientific Notation
Main IDEA
More than 425 million pounds of gold
has been discovered in the world. If all
this gold were in one place, it would
form a cube seven stories on each side.
Express numbers in
scientific notation and
in standard form.
New Vocabulary
1. Write 425 million in standard form.
scientific notation
2. Complete: 4.25 ×
?
= 425 million.
Math Online
glencoe.com
When you deal with very large numbers like 425,000,000, it can be
difficult to keep track of the zeros. You can express numbers such as this
in scientific notation by writing the number as the product of a factor
and a power of 10.
•Personal Tutor
•Self-Check Quiz
•Extra Examples
Scientific Notation
Key Concept
Words
A number is expressed in scientific notation when it is written
as the product of a factor and a power of 10. The factor must
be greater than or equal to 1 and less than 10.
Symbols
a × ​10 ​n​, where 1 ≤ a < 10 and n is an integer
Examples 425,000,000 = 4.25 × ​10 ​8​
Express Large Numbers in Standard Form
1 Express 2.16 × ​10 ​5​in standard form.
2.16 × 1​ 0 ​5​= 2.16 × 100,000​10 ​5​= 100,000
= 216,000
Move the decimal point 5 places to the right.
Express each number in standard form.
a. 7.6 × ​10 ​6​ b. 3.201 × ​10 ​4​ Scientific notation is also used to express
very small numbers. Study the pattern
of products at the right. Notice that
multiplying by a negative power of 10
moves the decimal point to the left the
same number of places as the absolute
value of the exponent.
LA2 Looking Ahead
1.25 × ​10 ​2​= 125
1.25 × ​10 ​1​= 12.5
1.25 × ​10 ​0​= 1.25
1.25 × ​10 ​–1​= 0.125
1.25 × ​10 ​–2​= 0.0125
1.25 × ​10 ​–3​= 0.00125
Express Small Numbers in Standard Form
2 Express 5.8 × 1​ 0 ​–3​in standard form.
Move the Decimal Point
The exponent tells you how
many places to move the
decimal point.
• If the exponent is
5.8 × ​10 ​–3​= 5.8 × 0.001 10–3 = 0.001
positive, move the
decimal point to
the right.
• If the exponent is
negative, move the
decimal point to the left.
= 0.0058
c. 4.7 × ​10 ​–5​ Move the decimal point 3 places to the left.
d. 9.0 × ​10 ​–4​ To write a number in scientific notation, place the decimal point after the
first nonzero digit. Then find the power of 10.
Express Numbers in Scientific Notation
Express each number in scientific notation.
3 1,457,000
1,457,000= 1.457 × 1,000,000 The decimal point moves 6 places to the left.
= 1.457 × ​10 ​6​
The exponent is positive.
4 0.00063
0.00063= 6.3 × 0.0001 The decimal point moves 4 places to the right.
= 6.3 × ​10 ​–4​
The exponent is negative.
e. 35,000 f. 0.00722 To compare numbers in scientific notation, first compare the exponents.
With positive numbers, any number with a greater exponent is greater.
If the exponents are the same, compare the factors.
Compare
Numbers in
Scientific Notation
5 OCEANS The Atlantic Ocean has an area of 3.18 × ​10 ​7​square miles.
The Pacific Ocean has an area of 6.4 × ​10 ​7​square miles. Which
ocean has the greater area?
Since both exponents are the same, compare the factors.
3.18 < 6.4 → 3.18 × ​10 ​7​< 6.4 × ​10 ​7​
So, the Pacific Ocean has the greater area.
g. Replace ● with <, >, or = to make 4.13 × ​10 ​–2​● 5.0 × ​10 ​–3​a true
sentence. Lesson 1 Scientific Notation LA3
Examples 1, 2
(pp. LA2–LA3)
Express each number in standard form.
1. 3.754 × ​10 ​5​ 3. 1.5 × ​10 ​–4​ Examples 3, 4
(p. LA3)
2. 8.34 × ​10 ​6​ 4. 2.68 × ​10 ​–3​ Express each number in scientific notation.
5. 4,510,000 6. 0.00673 7. 0.000092 8. 11,620,000 9. Physical science Light travels 300,000 kilometers per second. Write this
number in scientific notation. Example 5
(p. LA3)
10. Technology The distance between tracks on a
CD and DVD are shown in the table. Which disc
has the greater distance between tracks? Disc
CD
DVD
Distance (mm)
1.6 × ​10 ​-3​
7.4 × ​10 ​-4​
Replace each ● with <, >, or = to make a true sentence.
11. 2.3 × ​10 ​5​● 1.7 × ​10 ​5​ HOMEWORK
For
Exercises
13–22
23–32
33–38
HELP
See
Examples
1, 2
3, 4
5
12. 0.012 ● 1.4 × ​10 ​–1​ Express each number in standard form.
13. 6.1 × ​10 ​4​
14. 5.72 × ​10 ​6​
15. 3.3 × ​10 ​–1​
16. 5.68 × ​10 ​–3​
17. 9.014 × ​10 ​–2​
18. 1.399 × ​10 ​5​
19. 2.505 × ​10 ​3​
20. 7.4 × ​10 ​–5​
21. SPIDERS The diameter of a spider’s thread is 1.0 × 1
​ 0 ​–3​inch. Write this
number in standard form. 22. dinosaurs The Gigantosaurus dinosaur weighed about 1.4 × 1
​ 0 ​4​pounds.
Write this number in standard form. Express each number in scientific notation.
23. 499,000
24. 2,000,000
25. 0.006
26. 0.0125
27. 50,000,000
28. 39,560
29. 0.000078
30. 0.000425
31. CHESS The number of possible ways that a player can play the first four
moves in a chess game is 3 billion. Write this number in scientific
notation. 32. SCIENCE A particular parasite is approximately 0.025 inch long. Write this
number in scientific notation. SPORTS For Exercises 33 and 34, use the table.
Determine which category in each pair had a
greater amount of sales.
33. golf or tennis LA4 Looking Ahead
34. camping or golf Category
Camping
Golf
Tennis
Sales ($)
1.547 × ​10 ​9​
3.243 × ​10 ​9​
3.73 × ​10 ​8​
Source: National Sporting Goods Assoc.
Replace each ● with <, >, or = to make a true sentence.
35. 1.8 × ​10 ​3​● 1.9 × ​10 ​–1​ 37. 0.00701 ● 7.1 × ​10 ​–3​ 39. measurement The table at the right shows the values
36. 5.2 × ​10 ​2​● 5000 38. 6.49 × ​10 ​4​● 649 × ​10 ​2​ of different prefixes that are used in the metric system.
Write the units attometer, gigameter, kilometer,
nanometer, petameter, and picometer in order from
greatest to least measure.
40. NUMBER SENSE Write the product of 0.00004 and 0.0008
in scientific notation. Metric Measures
Prefix Meaning
atto
​10 ​-18​
giga
​10 ​9​
kilo
​10 ​3​
nano
​10 ​-9​
peta
​10 ​15​
pico
​10 ​-12​
41. NUMBER SENSE Order 6.1 × 1
​ 0 ​4​, 6100, 6.1 × ​10 ​–5​, 0.0061, and 6.1 × ​10 ​–2​
from least to greatest. PHYSICAL SCIENCE For Exercises 42 and 43,
use the table.
The table shows the maximum amounts of
lava in cubic meters per second that erupted
from four volcanoes.
42. How many times greater was the
Mount St. Helens eruption than the
Ngauruhoe eruption? Volcanic Eruptions
Eruption Rate
Volcano, Year
(​m ​3​/s)
Mount St. Helens, 2.0 × ​10 ​4​
1980
Ngauruhoe, 1975
2.0 × ​10 ​3​
Hekla, 1970
4.0 × ​10 ​3​
Agung, 1963
3.0 × ​10 ​4​
Source: University of Alaska
43. How many times greater was the Hekla
eruption than the Ngauruhoe eruption? Write each number in standard form.
44. (8 × ​10 ​0​) + (4 × ​10 ​–3​) + (3 × ​10 ​–5​) 45. (4 × ​10 ​4​) + (8 × ​10 ​3​) + (3 × ​10 ​2​) + (9 × ​10 ​1​) + (6 × ​10 ​0​) H.O.T. Problems
46. Challenge Convert the numbers in each expression to scientific notation.
Then evaluate the expression. Express in scientific notation and in
decimal notation.
(420,000)(0.015)
0.025
a.​ __
  
 
​ 
(0.078)(8.5)
0.16(250,000)
b.​ __
 
 ​ 
47. REASONING Which is a better estimate for the number of times per year
48. Open Ended Describe a real-life value or measure using numbers in
that a person blinks: 6.25 × ​10 ​–2​times or ​6.25 × 10 ​6​times? Explain your
reasoning. scientific notation and in standard form. 49.
WR ITING IN MATH Explain the relationship between a number in
standard form and the sign of the exponent when the number is written
in scientific notation. Lesson 1 Scientific Notation LA5