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Scientific Notation
Remember how?
Rules of Scientific Notation
4.23
coefficient
5
x 10
base exponent
 The coefficient must be greater than or equal to 1 and
less than 10.
 Must be base 10
 The exponent shows the number of places the decimal
must be moved to change the coefficient to a standard
number
 A standard number exists when the exponent is zero (0)
BAD EXAMPLES
 These are all BAD EXAMPLES of scientific notation.
DON’T DO THESE!!
Example
Why it’s incorrect
Corrected
0.34 x 107
Coefficient is not
between 1 and 10
3.4 x 106
25 x 10 -5
Coefficient is not
between 1 and 10
2.5 x 10-4
4.74 x 28
Not base 10
(we won’t be solving
for these)
4.74 x 256 = 1213.44 =
1.21344 x 103
Scientific Notation  Standard
 When going from scientific notation to standard, do
the following
 If the exponent is POSITIVE, move the decimal RIGHT
 Add place-holder zeroes as needed
 EX: 3.67 x 105  367000
 If the exponent is NEGATIVE, move the decimal LEFT
 Add place-holder zeroes as needed
 EX: 7.25 x 10-3  0.00725
Example
 Write 1.69 x 104 as a standard number
1 6 9 0 0 x 10
41032
Once you get to 100, you’re at the standard number.
When recording an answer, DO NOT put the 100. Leave it
out. Remember: x100 means x1
Example
 Write 4.23 x 10-3 as a standard number
0 0 0 4 2 3 x 10 -3-2-10
Once you get to 100, you’re at the standard number.
When recording an answer, DO NOT put the 100. Leave it
out. Remember: x100 means x1
Also, for neatness, it’s best to include the leading zero
before the decimal.
Standard  Scientific Notation
 When going from standard to scientific notation, do the
opposite as before, so:
 If you move the decimal LEFT, the exponent is POSITIVE
 EX: 8976  8.976 x 103
 If you move the decimal RIGHT, the exponent is NEGATIVE
 EX: 0.00058  5.8 x 10-4
Example
 Write 780374.2 in scientific notation.
7 8 0 3 7 4 2 x 10
7. Is a number between 1 and 10. We needed to move
the decimal 5 times to the left, so the exponent became
105.
5
1234
0
Example
 Write 0.006235 in scientific notation.
0 0 0 6 2 3 5 x
0
-3
-2
1
10
6 is a number between 1 and 10. We needed to move the
decimal 3 times to the right, so the exponent became 10-3.
Get rid of any leading zeroes.
Multiplying in Scientific Notation
 Example: 3.2 x 104 x 8.7 x 105
 Rules:
 MULTIPLY the coefficients together like usual
 3.2 x 8.7 = 27.84
 ADD the exponents together
 104 x 105 = 109
 Readjust for proper scientific notation, if needed
 27.84 x 109  2.784 x 1010
Multiplication Practice Problems
Problem
Work
Temp Answer
FINAL Answer
4.8 x 103 • 2.3 x 1012
4.8 • 2.3 = 11.04
103 • 1012 = 10(3 + 12) = 1015
11.04 x 1015
Can’t leave 11
1.104 x 1016
3.6 x 10-4 • 2.1 x 103
3.6 • 2.1 = 7.56
10-4 • 103 = 10(-4 + 3)=10-1
7.56 x 10-1
The 7 is ok
7.56 x 10-1
2.65 x 10-5 • 7.3 x 10-7 2.65 • 7.3 = 19.345
10-5 • 10-7 = 10(-5 + -7) = 10-12
19.345 x10-12
Can’t leave 19
1.9345 x 10-11
9.56 x 106 • 9.8 x 10-4 9.56 • 9.8 = 93.688
106 • 10-4 = 10(6 + -4) = 102
93.688 x102
Can’t leave 93
9.3688 x 103
2.1 • 7.22 = 15.162
15.162 x10-16
103 • 10-19 = 10(3 + -19)= 10-16 Can’t leave 15
1.5162 x 10-15
2.1 x 103 • 7.22 x 10-19
Dividing in Scientific Notation
 Example:
4.76 𝑥 107
8.3 𝑥 103
 DIVIDE the coefficients like usual (top divided by bottom)

4.76
8.3
= 0.573
 SUBTRACT the exponents (top # – bottom #)

107
103
= 104
 Readjust for proper scientific notation, if needed
 0.573 x 104  5.73 x 103
Division Practice Problems
Problem
3.31 𝑥 103
2.43 𝑥 108
6.7 𝑥 107
8.22 𝑥 103
3.0 𝑥 10−5
7.8 𝑥10−3
4.5 𝑥10−4
2.99 𝑥10−7
4 𝑥107
8.2 𝑥10−9
Work (coeff)
3.31
= 1.36
2.43
6.7
= 0.815
8.22
3.0
= 0.385
7.8
4.5
= 1.51
2.99
4
= 0.488
8.2
Work (exp)
103
= 10−5
8
10
107
= 104
3
10
10−5
−2
=
10
10−3
10−4
= 103
−7
10
107
= 1016
−9
10
Temp Answer
FINAL Answer
1.36 x 10-5
1.36 x 10-5
0.815 x 104
8.15 x 103
0.385 x 10-1
3.85 x 10-2
1.51 x 103
1.51 x 103
0.488 x 1016
4.88 x 1015
Scientific Method with Units
 Metric units have assigned values. When calculating with
those values, replace the unit with its value, then solve.
 The values are NOT the same as the ones for the factor
label conversions
 This is because they are absolute values, not comparisons to
the base unit.
Unit
Value
Sample
Equivalent (Scientific)
Equivalent (Standard)
kilo-
103
6.27 kg
6.27 x 103 g
6270 g
mega-
106
2.3 MHz
2.3 x 106 Hz
2300 000 Hz
nano-
10-9
7.4 nm
7.4 x 10-9 m
0.000 000 007 4 m
Practice Problems with Units
Problem
24 𝑘𝑔
2𝑔
230 𝑝𝑚 (𝑝𝑖𝑐𝑜)
52 𝑛𝑚 (𝑛𝑎𝑛𝑜)
Equivalent
24 𝑥 103
2
Work (coeff)
24
= 12
2
230
= 4.42
52
Work (exp)
103
= 103
0
10
2.3 ks • 16 s
230 𝑥 10−12
52 𝑥10−9
2.3 𝑥 103• 16
10−12
−3
=
10
10−9
2.3 • 16 = 36.8 103 • 100 = 103
0.4 kHz •
98 mHz
0.4 x 103 •
98 x 10-3
0.4 • 98 =
39.2
103 • 10-3 = 100
Answer
12 x 103 
1.2 x 104 g
4.42 x 10-3 m
(or 4.42 mm)
36.8 x 103 
3.68 x 104
39.2 x 100
3.92 x 101 Hz