2
Signed Numbers and
Functions
Powers of 10
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2.6
Scientific Notation
Copyright © Cengage Learning. All rights reserved.
Scientific Notation
Scientific Notation
Scientific notation is a method that is especially useful for
writing very large or very small numbers.
To write a number in scientific notation, write it as a product
of a number between 1 and 10 and a power of 10.
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Example 1
Write 226 in scientific notation.
226 = 2.26  102
Remember that 102 is a short way of writing 10  10 = 100.
Note that multiplying 2.26 by 100 gives 226.
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Scientific Notation
Writing a Decimal Number in Scientific Notation
To write a decimal number in scientific notation,
1. Reading from left to right, place a decimal point after the
first nonzero digit.
2. Place a caret (^) at the position of the original decimal
point.
3. If the decimal point is to the left of the caret, the
exponent of the power of 10 is the same as the number
of decimal places from the caret to the decimal point.
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Scientific Notation
4. If the decimal point is to the right of the caret, the
exponent of the power of 10 is the same as the negative
of the number of places from the caret to the decimal
point.
5. If the decimal point is already after the first nonzero digit,
the exponent of 10 is zero.
2.15 = 2.15  100
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Example 3
Write 2738 in scientific notation.
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Scientific Notation
Writing a Number in Scientific Notation in Decimal
Form
To change a number in scientific notation to decimal form,
1. Multiply the decimal part by the given positive power of
10 by moving the decimal point to the right the same
number of decimal places as indicated by the exponent
of 10. Supply zeros when needed.
2. Multiply the decimal part by the given negative power of
10 by moving the decimal point to the left the same
number of decimal places as indicated by the exponent
of 10. Supply zeros when needed.
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Example 5
Write 2.67  102 as a decimal.
2.67  102 = 267
Move the decimal point two places to the right,
since the exponent of 10 is +2.
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Scientific Notation
You may find it useful to note that a number in scientific
notation with
a. a positive exponent greater than 1 is greater than 10,
and
b. a negative exponent is between 0 and 1.
That is, a number in scientific notation with a positive
exponent represents a relatively large number.
A number in scientific notation with a negative exponent
represents a relatively small number.
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Scientific Notation
Scientific notation may be used to compare two positive
numbers expressed as decimals.
First, write both numbers in scientific notation.
The number having the greater power of 10 is the larger.
If the powers of 10 are equal, compare the parts of the
numbers that are between 1 and 10.
Scientific notation is especially helpful for multiplying and
dividing very large and very small numbers.
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Scientific Notation
To perform these operations, you must first know some
rules for exponents.
Multiplying Numbers in Scientific Notation
To multiply numbers in scientific notation, multiply the
decimals between 1 and 10.
Then add the exponents of the powers of 10.
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Example 10
Multiply (4.5  108)(5.2  10–14). Write the result in scientific
notation.
(4.5  108)(5.2  10–14) = (4.5)(5.2)  (108)(10–14)
= 23.4  10–6
= (2.34  101)  10–6
= 2.34  10–5
Note that 23.4  10–6 is not in scientific notation, because
23.4 is not between 1 and 10.
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Example 10
cont’d
To find this product using a calculator that accepts numbers
in scientific notation, use the following procedure.
Notes:
1. You may need to set your calculator in scientific
notation mode.
2. The
or
key is used to enter a negative number.
The product is 2.34  10–5.
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Scientific Notation
Dividing Numbers in Scientific Notation
To divide numbers in scientific notation, divide the decimals
between 1 and 10.
Then subtract the exponents of the powers of 10.
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Example 11
Divide
. Write the result in scientific notation.
Using a calculator, we have
The quotient is 3  104.
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Scientific Notation
Powers of Numbers in Scientific Notation
To find the power of a number in scientific notation, find the
power of the decimal between 1 and 10. Then multiply the
exponent of the power of 10 by this same power.
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Example 13
Find the power (4.5  106)2. Write the result in scientific
notation.
(4.5  106)2 = (4.5)2  (106)2
= 20.25  1012
Note that 20.25 is not between 1 and 10.
= (2.025  101)  1012
= 2.025  1013
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Example 13
cont’d
The result is 2.025  1013.
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