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PERFORMING
CALCULATIONS
IN SCIENTIFIC
NOTATION
ADDITION AND SUBTRACTION
Review:
Scientific notation expresses a
number in the form:
M x
Any number
between 1
and 10
n
10
n is an
integer
6
10
4 x
6
+ _______________
3 x 10
7 x 106
IF the exponents are
the same, we simply
add or subtract the
numbers in front and
bring the exponent
down unchanged.
106
4 x
+ 3 x 105
If the exponents are
NOT the same, we
must move a decimal
to make them the
same.
Determine which of the numbers has the smaller exponent.
1. Change this number by moving the decimal place to the
left and raising the exponent, until the exponents of both
numbers agree. Note that this will take the lesser number
out of standard form.
2. Add or subtract the coefficients as needed to get the new
coefficient.
3. The exponent will be the exponent that both numbers
share.
4. Put the number in standard form.
6
10
6
10
4.00 x
4.00 x
6
5
+ .30 x 10
+ 3.00 x 10
Move the decimal on the smaller
number to the left and raise the
exponent !
Note: This will take the lesser number out of standard form.
6
10
6
10
4.00 x
4.00 x
6
5
+ .30 x 10
+ 3.00 x 10
6
4.30 x 10
Add or subtract the coefficients
as needed to get the new
coefficient.
The exponent will be the exponent
that both numbers share.
Make sure your final answer is
in scientific notation. If it is
not, convert is to scientific
notation.!
A Problem for you…
-6
10
2.37 x
-4
+ 3.48 x 10
Solution…
-6
002.37
2.37 x 10
-4
+ 3.48 x 10
Solution…
-4
0.0237 x 10
-4
+ 3.48
x 10
-4
3.5037 x 10
PERFORMING
CALCULATIONS
IN SCIENTIFIC
NOTATION
MULTIPLYING AND DIVIDING
Rule for Multiplication
When multiplying with scientific notation:
1.Multiply the coefficients together.
2.Add the exponents.
3.The base will remain 10.
(2 x 103) • (3 x 105) =
6 x 108
(4.6x108) (5.8x106) =26.68x1014
Notice: What is wrong with this example?
Although the answer is correct, the
number is not in scientific notation.
To finish the problem, move the decimal one
space left and increase the exponent by
one.
26.68x1014 = 2.668x1015
((9.2
x 105) x (2.3 x 107) =
21.16 x 1012 =
2.116 x 1013
(3.2 x 10-5) x (1.5 x 10-3) =
4.8 • 10-8
Rule for Division
When dividing with scientific notation
1.Divide the coefficients
2.Subtract the exponents.
3.The base will remain 10.
(8 • 106) ÷ (2 • 103) =
4 x 103
Please multiply the following numbers.
(5.76 x 102) x (4.55 x 10-4) =
(3 x 105) x (7 x 104) =
(5.63 x 108) x (2 x 100) =
(4.55 x 10-14) x (3.77 x 1011) =
(8.2 x10-6) x (9.4 x 10-3) =
Please multiply the following numbers.
(5.76 x 102) x (4.55 x 10-4) = 2.62 x 10-1
(3 x 105) x (7 x 104) = 2.1 x 1010
(5.63 x 108) x (2 x 100) = 1.13 x 109
(4.55 x 10-14) x (3.77 x 1011) = 1.72 x 10-2
(8.2 x10-6) x (9.4 x 10-3) = 7.71 x 10-8
Please divide the following numbers.
1. (5.76 x 102) / (4.55 x 10-4) =
2. (3 x 105) / (7 x 104) =
3. (5.63 x 108) / (2) =
4. (8.2 x 10-6) / (9.4 x 10-3) =
5. (4.55 x 10-14) / (3.77 x 1011) =
Please divide the following numbers.
1. (5.76 x 102) / (4.55 x 10-4) = 1.27 x 106
2. (3 x 105) / (7 x 104) = 4.3 x 100 = 4.3
3. (5.63 x 108) / (2 x 100) = 2.82 x 108
4. (8.2 x 10-6) / (9.4 x 10-3) = 8.7 x 10-4
5. (4.55 x 10-14) / (3.77 x 1011) = 1.2 x 10-25
PERFORMING
CALCULATIONS
IN SCIENTIFIC
NOTATION
Raising Numbers in Scientific
Notation To A Power
(5 X 104)2 =
(5 X 104) X (5 X 104) =
(5 X 5) X (104 X 104) =
(25) X 108 = 2.5 X 109
Try These:
1. (3.45 X
1010)2
1.19 X 1021
2. (4 X 10-5)2
1.6 X 10-9
3. (9.81 X 1021)2
9.624 X 1043
1. (3.45 X 1010)2 = (3.45 X 3.45) X (1010 X 1010) = (11.9) X
(1020) = 1.19 X 1021
2. (4 X 10-5)2 = (4 X 4) X (10-5 X 10-5) = (16) X (10-10) = 1.6 X
10-9
3. (9.81 X 1021)2 = (9.81 X 9.81) X (1021 X 1021) = (96.24) X
(1042) =
9.624 X 1043
Changing from Standard
Notation to Scientific Notation
Ex. 6800
6800
1. Move decimal to get
a single digit # and
count places moved
3 2 1
68 x 10
2. Answer is a single
digit number times
the power of ten of
places moved.
3
Ex. 4.5 x 10 -3
00045
3 2 1
Changing from Scientific
Notation to Standard Notation
1. Move decimal the same
number of places as the
exponent of 10.
(Right if Pos. Left if Neg.)
9.54x107 miles
If the decimal is moved left the power is positive.
If the decimal is moved right the power is negative.
1.86x107 miles
per second
What is Scientific Notation
(3 x 104)(7 x 10–5)
Multiply two numbers
in Scientific Notation
= (3 x 7)(10 4 x 10–5)
1.
= 21 x 10-1
2.
3.
4.
= 2.1 x 10 0
or 2.1
Put #’s in ( )’s Put
base 10’s in ( )’s
Multiply numbers
Add exponents of 10.
Move decimal to put
Answer in Scientific
Notation
A number expressed in scientific notation is
expressed as a decimal number between 1 and 10
multiplied by a power of 10 e( g, 7000 = 7 x 103 or
0.0000019 = 1.9 x 10 -6)
Why do we use it?
It’s a shorthand way of writing very large or very
small numbers used in science and math and
anywhere we have to work with very large or very
small numbers.
2.0 x 10 2 + 3.0 x 103
6.20 x 10–5
8.0 x 103
6.20
8.0
= 0.775 x
10-5
103
10 -8
= 7.75 x 10–9
DIVIDE USING SCIENTIFIC
NOTATION
.2 x 10 3 + 3.0 x 103
= .2+3 x 103
= 3.2 x
1.
2.
Scientific
Notation
Makes
These
Numbers
Easy
Divide the #’s &
Divide the powers of ten
(subtract the exponents)
Put Answer in Scientific
Notation
103
Addition and subtraction
Scientific Notation
1. Make exponents of 10 the same
2. Add 0.2 + 3 and keep the 103 intact
The key to adding or subtracting numbers
in Scientific Notation is to make sure the
exponents are the same.
2.0 x 10 7 - 6.3 x 105
2.0 x 10 7 -.063 x 10 7
= 2.0-.063 x 10 7
= 1.937 x 10 7
1. Make exponents of 10 the same
2. Subtract 2.0 - .063 and
keep the 107 intact