Transcript Negative Exponents

Chapter 12
Exponents and
Polynomials
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12.2
Negative Exponents and
Scientific Notation
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Negative Exponents
Using the quotient rule,
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
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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
n
a  n
a
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Example
Simplify by writing each result using positive exponents
only.
1
1
a. 3  2 
3
9
1
7
b. x  7
x
2
2
c. 2x  4
x
4
Helpful Hint
Don’t forget that since
there are no parentheses,
x is the base for the
exponent –4.
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Example
Simplify by write each result using positive exponents
only.
a.  x 3   13
x
2
b.  3
c. (3)
2
1
1
 2 
9
3
1
1

2 
(3)
9
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Example
Simplify by writing each of the following expressions
with positive exponents.
a.
b.
1
x 3
x 2
y 4
3
1
x
 x3


1
1
3
x
1
4
2
y
 x  2
1
x
y4
(Note that to convert a power with a
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.)
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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
Power of a quotient
an
a
   n , c0
c
c
Quotient rule for exponents
am
mn

a
, a0
n
a
Zero exponent a0 = 1, a ≠ 0
1
n
a

, a0
Negative exponent
n
a
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Example
Simplify by writing the following expression with
positive exponents.
3 a b 
  4 7 3 
3 a b 
2
3 1
34 a14b 2
 8 6 6
3ab
2
3 a b
2



3 1 2
4
3
 ab
7 3 2
3

2 2

a

3 2
b
2
3   a  b 
4 2
 348 a146b26  34 a8b4
7 2
3 2
34 a 6 b 2
 8 14 6
3a b
8
a
a
 4 4 
4
81
b
3b
8
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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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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.
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Example
Write each of the following in scientific notation.
a.
4700
Move the decimal 3 places to the left, so that the new
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
b.
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
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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.
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Example
Write each of the following in standard notation.
a.
5.2738  103
Since the exponent is a positive 3, we move the decimal 3
places to the right.
5.2738  103 = 5273.8
b.
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
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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.
a.
(7.3  10-2)(8.1  105) = (7.3 · 8.1)  (10-2 · 105)
= 59.13  103
= 59,130
1.2 10 4 1.2 10 4
5

0
.
3

10


 0.000003
b.
9
9
4 10
4 10
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