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Copyright © 2012 Pearson Education, Inc.
Slide 7- 1
1.4
Exponential
and Scientific
Notation
■
■
■
■
■
The Product and Quotient Rules
The Zero Exponent
Negative Integers as Exponents
Simplifying (am)n
Raising a Product or a Quotient to
a Power
■ Scientific Notation
■ Significant Digits and Rounding
Copyright © 2012 Pearson Education, Inc.
Multiplying with Like Bases:
The Product Rule
For any number a and any positive integers
m and n,
a a  a
m
n
m n
.
(When multiplying powers, if the bases are
the same, keep the base and add the
exponents.)
Copyright © 2012 Pearson Education, Inc.
Slide 1- 3
Example
Multiply and simplify:
(a) z  z ; (b) (2x y )(7 x y )
8
3
3
7
2
11
Solution
(a) z  z  z  z
3 7
2 11
3
2
7
11
(b) (2 x y )(7 x y )  (2)  7  x  x  y  y
8
3
8 3
11
 14 x y
3 2
7 11
 14x y
5
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18
Slide 1- 4
Dividing with Like Bases:
The Quotient Rule
For any nonzero number a and any positive
integers m and n, m > n, a m
a
n
a
mn
.
(When dividing powers, if the bases are the
same, keep the base and subtract the
exponents of the denominator from the
exponent of the numerator.)
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Slide 1- 5
Example
Divide and simplify:
m15
18 x 6 y 8
(a) 7 ; (b)
m
9x2 y5
Solution
m15
(a) 7  m157  m8
m
18 x 6 y 8
6 2
85
(b)

2

x

y
9 x2 y5
4 3
 2x y
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Slide 1- 6
The Zero Exponent
For any nonzero real number a,
a0  1.
(Any nonzero number raised to the zero
power is 1. 00 is undefined.)
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Slide 1- 7
Example
Evaluate each of the following for y = 5:
(a) y ; (b)  2 y ;
0
(c) (2 y) .
0
0
Solution
(a) y  5  1
0
0
(b)  2 y  2  5  2
0
0
(c) (  2 y)  (2  5)  (10)  1
0
0
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0
Slide 1- 8
Integer Exponents
For any real number a that is nonzero and
any integer n,
a
n

1
a
.
n
(The numbers a-n and an are reciprocals of
each other.)
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Slide 1- 9
Example
Express using positive exponents and simplify if
possible.
(a) 12
2
Solution
6
(b) 2 x y
3
1
(c) 4
m
1
1
(a) 12  2 
12 144
3
 1  3 2y
6 3
(b) 2 x y  2  6  y  6
x
x 
1
(c) 4  m ( 4)  m4
m
2
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Slide 1- 10
Factors and Negative Exponents
For any nonzero real numbers a and b and
any integers m and n,
a
b
n
m

b
m
a
.
n
(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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Slide 1- 11
Example
Write an equivalent expression without negative
exponents:
51 x 6 y 3
.
4
w
Solution
51 x 6 y 3 w4 y 3
= 6
4
w
5x
Copyright © 2012 Pearson Education, Inc.
Slide 1- 12
The product and quotient rules apply for all integer
exponents.
Example
Simplify:
(a) x 2  x 13 ;
m 2
(b) 4 .
m
Solution
(a) x  x
2
2
13
x
2  ( 13)
x
11
1
 11
x
m
2  ( 4)
2
(b) 4  m
m
m
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Slide 1- 13
The Power Rule
For any real number a and any integers m
and n,
m n
mn
(a )  a .
(To raise a power to a power, multiply the
exponents.)
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Slide 1- 14
Example
Simplify:
2 4
3 8
(a) (x ) ; (b) (5 ) ;
5 16
(c) (m ) .
Solution
1
(a) (x )  x  x  8
x
3 8
38
24
(b) (5 )  5  5
2 4
5 16
(c) (m )
24
8
m
5( 16)
m
80
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Slide 1- 15
Raising a Product or a
Quotient to a Power
Raising a Product to a Power
For any integer n, and any real numbers a
and b for which (ab)n exists,
(ab)  a b .
n
n n
(To raise a product to a power, raise each
factor to that power.)
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Slide 1- 16
Example
Simplify:
3
4
5 3
(a) (3y) ; (b) (5x y ) .
Solution
(a) (3y)  3  y  81y
4
4
4
4
(b) (5x y )  5 ( x ) ( y )
3
5 3
3
3 3
5 3
 125x y
9
15
125 y15

x9
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Slide 1- 17
Raising a Quotient to a Power
For any integer n, and any real numbers a
and b for which a/b, an, and bn exist,
n
n
a
a
   n.
b
b
(To raise a quotient to a power, raise both the
numerator and denominator to that power.)
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Slide 1- 18
Example
Simplify:
2
5
 9
 xy 
(a)  4  ; (b)  2  .
 z 
y 
3
Solution
2
2
 9
9
81
(a)  4   4 2  8
y
 y  (y )
5
( xy 3 ) 5 x 5 y15
y15
 xy 
(b)  2   2 5  10  5 10
(z )
z
xz
 z 
3
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Slide 1- 19
Definitions and Properties of Exponents
The following summary assumes that no denominators are 0 and that
00 is not considered. For any integers m and n,
a1 = a
a0 = 1
a  n  a1n
1 as an exponent:
0 as an exponent:
Negative Exponents:
 

a n
b
n
mn
a n
b m
b n
a


bm
an
am  a  a
The Product Rule:
The Quotient Rule:
am
 a mn
n
a
The Power Rule:
(am)n = amn
Raising a product to a power:
(ab)n = anbn
n
Raising a quotient to a power:
an
a
   n
b
b
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Slide 1- 20
Scientific Notation
Scientific Notation
Scientific notation for a number is an
m
N

10
expression of the form
, where N is in
decimal notation, 1  N  10 and m is an
integer.
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Slide 1- 21
Converting
Decimal Notation
571,000,000
0.00000063
0.59
190
Scientific Notation
=
5.71  10
=
6.3 10
=
5.9 10
=
1.9 10
8
7
1
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2
Slide 1- 22
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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Slide 1- 23
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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Slide 1- 24
Significant Digits and Rounding
When two or more measurements written in
scientific notation are multiplied or divided, the
result should be rounded so that it has the same
number of significant digits as the measurement
with the fewest significant digits. Rounding
should be performed at the end of the calculation.
Copyright © 2012 Pearson Education, Inc.
Slide 1- 25
Significant Digits and Rounding
When two or more measurements written in
scientific notation are added or subtracted, the
result should be rounded so that it has as many
decimal places as the measurement with the
fewest decimal places.
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Slide 1- 26
Example
Multiply and write scientific notation for the
answer:
(2.7 10 )(3.15 10 ).
8
3
Solution
(2.7 108 )(3.15 103 )   2.7  3.15 108 103 
 8.505 1011
rounded to 2 significant digits:
 8.5 1011
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Slide 1- 27
Example
Divide and write scientific notation for the
9
6.2

10
answer:
.
8
8.0 10
Solution
6.2  109 6.2 109

 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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Slide 1- 28
Example
Use a graphing calculator to calculate (–5.2)3.
Solution
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Slide 1- 29
Example
Use a graphing calculator to calculate
(6.2 × 103) (3.1 × 10–12).
Solution
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Slide 1- 30