Transcript BEI06_ppt_0402

Chapter 4
Polynomials
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4.2
Negative Exponents and
Scientific Notation
• Negative Integers as Exponents
• Scientific Notation
• Multiplying and Dividing Using Scientific
Notation
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Negative Exponents
For any real number a that is nonzero and
any integer n, a  n  1n .
a
(The numbers a-n and an are reciprocals of
each other.)
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Example
Express using positive exponents, and, if possible,
simplify.
a) m5
b) 52
c) (4)2
d) xy1
Solution
1
5
a) m = 5
m
b)
52
1
1
= 2 
25
5
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Example
c)
(4)2 =
d) xy1
1
1
1


2
(4)(4) 16
(4)
 1
1 x
= x 1   x  
 y y
y 
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Example
Simplify. Do not use negative exponents in the
answer.
a) w5  w3
b) (x4)3
c) (3a2b4)3
5
d)
a
a 6
1
e) 9
b
7
w
f)
6
z
Solution
a) w5  w3  w5 ( 3)  w2
b) (x4)3 = x(4)(3) = x12
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Example
c) (3a2b4)3 = 33(a2)3(b4)3
5
6
27a
= 27 a6b12 = 12
b
a
5 ( 6)
1
d) 6  a
a a
a
e) 1
 ( 9 )
9
b
b
9
b
7
6
w
1
1
z
7
6
f)
 w  6  7  z  7
6
z
z
w
w
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Factors and Negative Exponents
For any nonzero real numbers a and b and
any integers m and n,
a  n bm
 n.
m
b
a
(A factor can be moved to the other side of
the fraction bar if the sign of the exponent
is changed.)
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Example
6

20
x
Simplify.
4 y 3 z 4
Solution
We can move the negative factors to the other side
of the fraction bar if we change the sign of each
exponent.
6
20 x
5 z
 3 6
3 4
4y z
y x
4
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Reciprocals and Negative
Exponents
For any nonzero real numbers a and b and any
integer n,
a
 
b
n
n
b
  .
a
(Any base to a power is equal to the reciprocal of
the base raised to the opposite power.)
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Example
Simplify:  a 
4
2
 
 3b 
Solution
2
a 
 3b 
   4 
a 
 3b 
4
2
(3b) 2
 4 2
(a )
2
2
2
3 b
9b
 8  8
a
a
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Scientific Notation
Scientific notation for a number is an
expression of the type
N × 10m,
where N is at least 1 but less than 10 (that is,
1 ≤ N < 10), N is expressed in decimal
notation, and m is an integer.
Note that when m is positive the decimal point moves right m
places in decimal notation. When m is negative, the decimal point
moves left |m| places.
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Example
Convert to decimal notation:
a) 3.842  106
b) 5.3  107
Solution
a) Since the exponent is positive, the decimal
point moves right 6 places.
3.842000. 3.842  106 = 3,842,000
b) Since the exponent is negative, the decimal
point moves left 7 places.
0.0000005.3 5.3  107 = 0.00000053
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Example
Write in scientific notation:
a) 94,000
b) 0.0423
Solution
a) We need to find m such that 94,000
= 9.4  10m. This requires moving the decimal
point 4 places to the right.
94,000 = 9.4  104
b) To change 4.23 to 0.0423 we move the
decimal point 2 places to the left.
0.0423 = 4.23  102
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Multiplying and Dividing Using Scientific
Notation
Products and quotients of numbers written in
scientific notation are found using the rules for
exponents.
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Example
Simplify: (1.7  108)(2.2  105)
Solution (1.7  108)(2.2  105)
= 1.7  2.2  108  105
= 3.74  108 +(5)
= 3.74  103
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Example
Simplify. (6.2  109)  (8.0  108)
Solution
9
9
6.2

10
6.2
10
9
8
(6.2  10 )  (8.0  10 ) =

 8
8
8.0 10
8.0  10
 0.775  1017
 7.75 101 1017
 7.75  1018
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