Scientific Notation
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Transcript Scientific Notation
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