Scientific Notation
Scientific Notation
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A way of succinctly writing very large or very small numbers
Always has one non-zero digit in front of the decimal point
Requires moving decimal points and using powers of ten
Exponent rules make calculations easier
To Put a Number into Scientific Notation
 Move the decimal point until it is behind exactly one non-zero digit
o When moving the decimal to the left (  ), the exponent will be positive
 Moving the decimal to the left makes the number smaller. We’ll need to multiply by a
large number to maintain equality
o When moving the decimal to the right (  ), the exponent will be negative
 Moving the decimal to the right makes the number bigger. We’ll need to multiply by a
small number to maintain equality
 Only write non-zero digits, followed by x10 raised to the power of however many places you moved the
decimal point, with the sign as indicated by the direction of movement
 9,203,400,000,000 = 9.2034 x 1012
To Take a Number out of Scientific Notation
 Undo the steps above, working backwards
 Look at the exponent on the 10
o If the exponent is positive, it means the decimal was moved to the left. To undo this, move it to
the right however many places are indicated by the number itself
o If the exponent is negative, it means the decimal was moved to the right. To undo this, move it to
the left however many places are indicated by the number itself
 Fill in zeros as necessary to maintain the correct placement of the decimal point
 8.125 x 10-10 = 0.000 000 000 812 5
Powers of ten
1
10
100
1 , 000
10 , 000
100 , 000
1 , 000 , 000
1 , 000 , 000 , 000
1 , 000 , 000 , 000 , 000
1 , 000 , 000 , 000 , 000 , 000
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-
=
=
=
=
=
=
=
=
=
=
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 9
10 12
10 15
10 0 =
10 -1 =
10 -2 =
10 -3 =
10 -4 =
10 -5 =
10 -6 =
10 -9 =
10 -12 =
10 -15 =
1
0.
0.
0.
0.
0.
0.
0.
0.
0.
1
01
001
000 1
000 01
000 001
000 000 001
000 000 000 001
000 000 000 000 001
Positive powers of ten represent how many zeros are behind the 1
o 105 means 1 with 5 zeros behind it, or 100,000
Negative powers of ten, because the decimal also needs to be moved behind the 1 itself, contain one less
zero than indicated by the power
o 10-5 means 1 with a decimal and four zeros in front of it, or 0.00001
Always verify by counting the number of places the decimal has moved, not simply zeros
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Scientific Notation
Practice

Put 7,000 into scientific notation
- If there is no decimal point written, it’s behind the ones digit
- Count the number of places the decimal point must be moved so there is exactly one nonzero digit in front of it
 7000.
- Three places to the left means x 103
 7 x 103

Put 0.00000524 into scientific notation
- Count the number of places the decimal point must be moved so there is exactly one nonzero digit in front of it
 0.00000524
- Six places to the right means x 10-6
 5.24 x 10-6

Take 3.01 x 104 out of scientific notation
- Look at the exponent on the 10
 Positive 4 means the decimal had been moved four places to the left. Undo this by
moving it four places to the right
 3.01
- Fill in zeros as needed to maintain the decimal in this location
 30100

Take 8.74 x 10-2 out of scientific notation
- Look at the exponent on the 10
 Negative 2 means the decimal had been moved two places to the right. Undo this by
moving it two places to the left
 8.74
- Fill in zeros as needed to maintain the decimal in this location
 0.0874
Error analysis
 Put 98.1 x 103 into proper scientific notation
- There can only be one digit in front of the decimal point
- Because we’re moving the decimal point to the left, we increase the exponent
 9.81 x 104

Put 0.035 x 10-6 into proper scientific notation
- There needs to be one non-zero digit in front of the decimal point
- Because we’re moving the decimal point to the right, we decrease the exponent
 3.5 x 10-8
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Scientific Notation
Multiplying and Dividing in Scientific Notation
 Multiply (or divide) the digits
 Multiply (or divide) the powers of ten using the rules of exponents
o Add the exponents when multiplying
o Subtract the exponents when dividing
 Combine the expressions, recombining the powers of ten
o Recombine with multiplication, regardless of what operation was used in the problem
Practice
Multiply 7 x 103 by 5.24 x 10-6
 Multiply the digits
- 5.24 x 7 = 36.68
- Move the decimal point so there is one digit in front of the decimal point
- 3.668 x 101
 Multiply the powers of ten using the rules of exponents (add the exponents)
- 103 x 10-6 = 103 + -6 = 10-3
 Combine the expressions, multiplying the powers of ten
- 3.668 x 101 x 10-3
- 3.668 x 10-2
o This can be done all at once as follows
 (7 x 103) x (5.24 x 10-6)
 (7 x 5.24) x (103 x 10-6)
 36.68 x 10-3
 3.668 x 10-2
Divide 6.48 x 10-4 by 8 x 106
 Divide the digits
- 6.48 ÷ 8 = 0.81
- Move the decimal point so there is one digit in front of the decimal point
- 8.1 x 10-1
 Divide the powers of ten using the rules of exponents (subtract the exponents)
- 10-4 ÷ 106 = 10-4 - 6 = 10-10
- Combine the expressions, multiplying the powers of ten
- 8.1 x 10-1 x 10-10
- 8.1 x 10-11
o This can be done all at once as follows
 (6.48 ÷ 10-4) x (8 x 106)
 (6.48 ÷ 8) x (10-4 ÷ 106)
 0.81 x 10-10
 8.1 x 10-11
We can verify these longhand
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7000 x 0.00000524 = 0.03668 = 3.668 x 10-2
0.000648 ÷ 8,000,000 = 0.000 000 000 081 = 8.1 x 10-11
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