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Adding/Subtracting/Multiplying
/Dividing Numbers in Scientific
Notation
How wide is our universe?
210,000,000,000,000,000,000,0
00 miles
(22 zeros)
This number is written in decimal
notation. When numbers get this
large, it is easier to write them in
scientific notation.
Scientific Notation
A number is expressed in
scientific notation when it is
in the form
a x 10n
where a is between 1 and 10
and n is an integer
An easy way to
remember this is:
• If an exponent is positive, the
number gets larger, so move the
decimal to the right.
• If an exponent is negative, the
number gets smaller, so move the
decimal to the left.
When changing from Standard Notation
to Scientific Notation:
4) See if the original number is greater
than or less than one.
– If the number is greater than one, the
exponent will be positive.
348943 = 3.489 x 105
– If the number is less than one, the
exponent will be negative.
.0000000672 = 6.72 x 10-8
Write the width of the
universe in scientific notation.
210,000,000,000,000,000,000,000
miles
Where is the decimal point now?
After the last zero.
Where would you put the decimal to
make this number be between 1
and 10?
Between the 2 and the 1
2.10,000,000,000,000,000,
000,000.
How many decimal places did you move
the decimal?
23
When the original number is more
than 1, the exponent is positive.
The answer in scientific notation is
2.1 x 1023
Write 28750.9 in scientific
notation.
1.
2.
3.
4.
2.87509 x 10-5
2.87509 x 10-4
2.87509 x 104
2.87509 x 105
-4
10
2) Express 1.8 x
in decimal
notation.
0.00018
3) Express 4.58 x 106 in decimal
notation.
4,580,000
Try changing these numbers from
Scientific Notation to Standard
Notation:
1) 9.678 x 104
96780
2) 7.4521 x 10-3
.0074521
3) 8.513904567 x 107
85139045.67
4) 4.09748 x 10-5
.0000409748
Write in PROPER scientific notation.
(Notice the number is not between 1 and
10)
8) 234.6 x 109
2.346 x 1011
9) 0.0642 x 104
6.42 x 10
2
Adding/Subtracting when
Exponents are Equal
• When the exponents are the same
for all the numbers you are working
with, add/subtract the base numbers
then simply put the given exponent on
the 10.
General Formulas
• (N X 10x) + (M X 10x) = (N + M) X 10x
• (N X 10y) - (M X 10y) = (N-M) X 10y
Example 1
• Given: 2.56 X 103 + 6.964 X 103
• Add: 2.56 + 6.964 = 9.524
• Answer: 9.524 X 103
Example 2
• Given: 9.49 X 105 – 4.863 X 105
• Subtract: 9.49 – 4.863 = 4.627
• Answer: 4.627 X 105
Adding With the Same
Exponent
• (3.45 x 103) + (6.11 x 103)
• 3.45 + 6.11 = 9.56
• 9.56 x 103
Subtracting With the
Same Exponent
• (8.96 x 107) – (3.41 x 107)
• 8.96 – 3.41 = 5.55
• 5.55 x 107
Adding/Subtracting when
the Exponents are
Different
• When adding or subtracting numbers
in scientific notation, the exponents
must be the same.
• If they are different, you must move
the decimal either right or left so
that they will have the same
exponent.
Moving the Decimal
• For each move of the decimal to the
right you have to add -1 to the
exponent.
• For each move of the decimal to the
left you have to add +1 to the
exponent.
Continued…
• It does not matter which number you
decide to move the decimal on, but
remember that in the end both
numbers have to have the same
exponent on the 10.
Example 1
• Given: 2.46 X 106 + 3.476 X 103
• Shift decimal 3 places to the left for
103.
• Move: .003476 X 103+3
• Add: 2.46 X 106 + .003476 X 106
• Answer: 2.463 X 106
Example 2
• Given: 5.762 X 103 – 2.65 X 10-1
• Shift decimal 4 places to the right
for 10-1.
• Move: .000265 X 10(-1+4)
• Subtract: 5.762 X 103-.000265 X 103
• Answer: 5.762 X 103
•
•
•
•
(4.12 x 106) + (3.94 x 104)
(412 x 104) + (3.94 x 104)
412 + 3.94 = 415.94
415.94 x 104
• Express in proper form:
4.15 x 106
Subtracting With
Different Exponents
•
•
•
•
(4.23 x 103) – (9.56 x 102)
(42.3 x 102) – (9.56 x 102)
42.3 – 9.56 = 32.74
32.74 x 102
• Express in proper form: 3.27 x 103
Multiplying…
• The general format for multiplying is
as follows…
• (N x 10x)(M x 10y) = (N)(M) x 10x+y
• First multiply the N and M numbers
together and express an answer.
• Secondly multiply the exponential
parts together by adding the
exponents together.
Multiplying…
• Finally multiply the two results for
the final answer.
• (2.41 x 104)(3.09 x 102)
– 2.41 x 3.09 = 7.45
–4+2=6
– 7.45 x 106
7) evaluate
(3,600,000,000)(23).
The answer in scientific
notation is
8.28 x 10 10
The answer in decimal notation
is
82,800,000,000
6) evaluate
(0.0042)(330,000).
The answer in decimal notation
is
1386
The answer in scientific
notation is
1.386 x 103
Write (2.8 x 103)(5.1 x 10-7) in
scientific notation.
1.
2.
3.
4.
14.28 x 10-4
1.428 x 10-3
14.28 x 1010
1.428 x 1011
Now it’s your turn.
• Use the link below to practice
multiplying numbers in scientific
notation.
• Multiplying in Scientific Notation
Dividing…
• The general format for dividing is as
follows…
• (N x 10x)/(M x 10y) = (N/M) x 10x-y
• First divide the N number by the M
number and express as an answer.
• Secondly divide the exponential parts
by subtracting the exponent from
the exponent in the upper number.
Dividing…
• Finally divide the two results
together to get the final answer.
• (4.89 x 107)/(2.74 x 104)
• 4.89 / 2.74 = 1.78
• 7–4=3
• 1.78 x 103
5) evaluate:
7.2 x 10-9
2
1.2 x 10
:
The answer in scientific
notation is
6 x 10 -11
The answer in decimal notation
is
0.00000000006
4) Evaluate:
4.5 x 10-5
1.6 x 10-2
0.0028125
Write in scientific notation.
2.8125 x 10-3
Now it’s your turn.
• Use the link below to practice
dividing numbers in scientific
notation.
• Dividing in Scientific Notation
Practice Worksheet
• Practice Adding and Subtracting in
Scientific Notation
• Answers to Worksheet
Links for more
information and practice
• Addition and Subtraction with
Scientific Notation
• Problem Solving--Scientific Notation
• Scientific Notation
Quiz Time!!!
• Below is a set of links for a quiz on
adding and subtracting numbers in
scientific notation, and there is a link
to get the answers to the quiz.
• Adding and Subtracting Quiz
• Answers to Quiz