Transcript 3.2

3.2
Negative Exponents and
Scientific Notation
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Negative Exponents
Using the quotient rule from section 3.1,
4
x
46
2
x x
6
x
x0
But what does x -2 mean?
x
x x x x
1
1


 2
6
x
x x x x x x x x x
4
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Negative Exponents
In order to extend the quotient rule to cases
where the difference of the exponents would
give us a negative number we define negative
exponents as follows.
If a is a real number other than 0, and n is an
integer, then
1
a
n

a
n
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Simplifying Expressions
Example
Simplify. Write each result using positive exponents only.
1
1
3  2 
3
9
1
7
x  7
x
2
2 x 4 
2
x4
Helpful Hint
Don’t forget that since
there are no parentheses,
x is the base for the
exponent –4.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Simplifying Expressions
Example
Simplify. Write each result using positive exponents only.
x
3

 32  
(3)2 
1
x3
1
1


32
9
1
1

2
(3)
9
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Simplifying Expressions
Example
Simplify by writing each of the following expressions
with positive exponents.
1)
2)
1
x 3
x 2
y 4
3
1
x  x3


1
1
x3
(Note that to convert a power with a
1
4
2
y
 x  2
1
x
y4
negative exponent to one with a positive
exponent, you simply switch the power
from the numerator to the denominator, or
vice versa, and switch the exponent to its
opposite value.)
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Summary of Exponent Rules
If m and n are integers and a and b are real numbers, then:
Product Rule for exponents am · an = am+n
Power Rule for exponents (am)n = amn
Power of a Product (ab)n = an · bn
n
an
a
Power of a Quotient    n , b  0
b b
Quotient Rule for exponents
am
mn

a
, a0
n
a
Zero exponent a0 = 1, a ≠ 0
1
Negative exponent a  n  n , a  0
a
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Simplifying Expressions
Simplify by writing the following expression with
positive exponents or calculating.
3 a b
3  a  b 
 3 ab 


 34 a 7 b 3   34 a 7 b 3 2  4 2 7 2 3 2

 3  a  b 
2
2
3
2
3
2
Power of a quotient rule
4
14
2
3 a b
3 a b
 8 14 6  8 6 6
3 a b
3a b
4
6
2
Power rule for exponents
2 2
3 2
2
Power of a product rule
8
a
a
4 8 4
48 146 26


3 a b 3 a b
4 4
3 b 81b 4
8
Quotient rule for exponents
Negative exponents
Negative exponents
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Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Scientific Notation
In many fields of science we encounter very large
or very small numbers. Scientific notation is a
convenient shorthand for expressing these types of
numbers.
A positive number is written in scientific notation
if it is written as the product of a number a, where
1 ≤ a < 10, and an integer power r of 10:
a  10r
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Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Scientific Notation
To Write a Number in Scientific Notation
Step 1: Move the decimal point in the original number so
that the new number has a value between 1 and 10
Step 2: Count the number of decimal places the decimal
point is moved in Step 1. If the original number is 10
or greater, the count is positive. If the original
number is less than 1, the count is negative.
Step 3: Multiply the new number in Step 1 by 10 raised to an
exponent equal to the count found in Step 2.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Scientific Notation
Example
Write each of the following in scientific notation.
Move the decimal 3 places to the left, so that the new
1) 4700
number has a value between 1 and 10.
Since we moved the decimal 3 places, and the original
number was > 10, our count is positive 3.
4700 = 4.7  103
2)
0.00047
Move the decimal 4 places to the right, so that the new
number has a value between 1 and 10.
Since we moved the decimal 4 places, and the original
number was < 1, our count is negative 4.
0.00047 = 4.7  10-4
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Scientific Notation
In general, to write a scientific notation number in
standard form, move the decimal point the same
number of spaces as the exponent on 10. If the
exponent is positive, move the decimal point to the
right. If the exponent is negative, move the decimal
point to the left.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Scientific Notation
Example
Write each of the following in standard notation.
1) 5.2738  103
Since the exponent is a positive 3, we move the decimal 3
places to the right.
5.2738  103 = 5273.8
2)
6.45  10-5
Since the exponent is a negative 5, we move the decimal
5 places to the left.
00006.45  10-5 = 0.0000645
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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Operations with Scientific Notation
Multiplying and dividing with numbers written in scientific
notation involves using properties of exponents.
Example
Perform the following operations.
1) (7.3  10-2)(8.1  105) = (7.3 · 8.1)  (10-2 · 105)
= 59.13  103
= 59,130
1.2  104 1.2 104
5

0
.
3

10


 0.000003
2)
9
9
4  10
4 10
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Prealgebra & Introductory Algebra, 3ed
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