Powers of Ten and Scientific Notation
Large numbers such as the speed of light and the
distance from the Earth to the Sun are often
represented using an abbreviated form.
In science, we refer to the speed of light as:
3 × 108 m/s.
This is the abbreviated form of 300 000 000 metres per second.
Powers of 10 are used to estimate and reduce the size of large numbers to make them
easier to work with.
A power of 10 is a power with a base of 10.
10
The exponent can be
any number
Power of 10
Repeated
Multiplication
Answer
102
10 × 10
100
The exponent is 2.
There are 2 zeros in the answer.
For example:
3
• 10 × 10 × 10 = 1000, and can also be represented as 10 .
5
• 10 × 10 × 10 × 10 × 10 = 100 000, and can also be represented as 10 .
Multiples of 10 can be represented as powers of 10.
Examples
A) 7000 is 7 × 1000 and can be represented as 7 × 103.
B) 20 000 is 2 × 10 000 and can be represented as 2 × 104.
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Scientific Notation
Scientific notation is a way of writing numbers in powers of 10 and is used when
communicating large numbers.
Example
The distance between Earth and the Sun is approximately 150 000 000 000 m.
This number is abbreviated as:
1.5 × 1011 m
This number represents a scientific notation.
Numbers in scientific notation look like this example.
In general, scientific notation looks as follows.
There may be one or more
digits to the right of the decimal
exponent
decimal point
×
A number that is at least
1, but less than 10
Example
multiplied by
10
a power of 10
1.275 × 103 (original number = 1275)
Notice that in representing numbers in scientific notation:
•
•
there is only one number in front of the decimal point
there may be several numbers to the right of the decimal point.
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Writing Numbers in Scientific Notation Using Repeated Multiplication
Example
What is 1275 in scientific notation?
127.5 × 10
= 1275
12.75 × 10 × 10
= 12.75
1.275 × 10 × 10 × 10 = 1.275
There are three 10’s being multiplied.
Therefore: 1275 = 1.275 × 103 in scientific notation
Writing Numbers in Scientific Notation by Moving the Decimal Point
This is a good mind math method. Some people call it a shortcut.
Example
What is 1275 in scientific notation?
•
In all whole numbers, the decimal point is to the right of the last
digit—we do not place it unless it is needed.
•
The decimal point is located behind 5.
1275.
•
Move the decimal point to the LEFT until there is only one number to
the left of the decimal point.
1275. Æ
•
127.5 Æ
12.75 Æ 1.275
Next, COUNT the number of times you moved the decimal
point to the left (3 times in the example). This number is your
exponent in the power of 10.
Therefore:
1 2 7 5.
=
1.275 × 103 in scientific notation
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Examples
A) What is 4000 in scientific notation?
4000
4 0 0 0.
=
4 × 103
Decimal moved 3 places.
B) What is 3 000 000 in scientific notation?
3 000 000
3 0 0 0 0 0 0.
=
3 × 106
Decimal moved 6 places.
C) What is 345 in scientific notation?
345
3 4 5. =
3.45 × 102
Decimal moved 2 places.
D) What is 12 065 in scientific notation rounded to the nearest tenth?
12 065
1 2 0 6 5. =
1.2065 × 104 =
1.2 × 104
after rounding
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Practice: Working with Powers of Ten
1. Do you notice a pattern between the number of zeros following the one, and the
exponent? Explain this pattern to your teacher or a classmate.
2. Write each number in repeated multiplication and as a power of 10.
Number
Repeated
Multiplication
Power of 10
1 000
103
10 000
100 000
10 × 10 × 10 × 10 × 10
1 000 000
3. Maddy was examining the population of cities in Alberta. She found the following
statistics:
Calgary – 876 519
Edmonton – 648 284
Grande Prairie – 35 962
Lethbridge – 68 712
Medicine Hat – 50 512
Red Deer – 68 308
Maddy created a chart in which she rounded each number and identified each
population as a power of 10.
Cities
Calgary
Edmonton
Grande Prairie
Lethbridge
Medicine Hat
Red Deer
Rounded Populations
(to the nearest 10 000)
880 000
650 000
40 000
70 000
50 000
70 000
Populations in
Powers of 10
8.8 × 106
6.5 × 104
4 × 104
7 × 104
5 × 105
7 × 104
Luckily, when Maddy rechecked her work before handing it in, she identified three
errors. Can you find and correct her three errors? (Hint: The errors are not in the
rounded numbers.)
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4. The chart below is a result of Census 2001. It shows the population of each of
Canada’s provinces and territories. Rewrite each population figure in scientific
notation. Round each number to one digit to the right of the decimal point.
Province/Territory
British Columbia
Alberta
Saskatchewan
Manitoba
Ontario
Quebec
New Brunswick
Nova Scotia
Prince Edward Island
Newfoundland
Yukon Territory
Northwest Territories
Nunavut
Population
Scientific Notation
3 907 738
2 974 807
978 933
1 119 583
11 410 046
7 237 479
729 498
908 007
135 294
512 930
28 674
37 360
26 745
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5. Rewrite each figure below in scientific notation. Round each number to one digit to
the right of the decimal point.
Number
Scientific Notation
14 532 739
27 466 570
39 774 129
47 453 982
73 283 465
96 374 102
137 845 923
177 299 374
187 378 069
566 734 292
1 456 292 349
3 744 239 129
12 373 460 595
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