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Chapter 4
Polynomials
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
4.1 Exponents and Their Properties
• Multiplying Powers with Like Bases
• Dividing Powers with Like Bases
• Zero as an Exponent
• Raising a Power to a Power
• Raising a Product or a Quotient to a Power
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-2
The Product Rule
For any number a and any positive integers
m and n,
a m  a n  a mn .
(To multiply powers with the same base,
keep the base and add the exponents.)
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
Example
Multiply and simplify each of the following. (Here
“simplify” means express the product as one base
to a power whenever possible.)
a) x3  x5
b) 62  67  63
c) (x + y)6(x + y)9
d) (w3z4)(w3z7)
Solution
a) x3  x5 = x3+5
= x8
Adding exponents
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-4
Example
b) 62  67  63
d) (w3z4)(w3z7)
c) (x + y)6(x + y)9
Solution
b) 62  67  63 = 62+7+3
= 612
c) (x + y)6(x + y)9 = (x + y)6+9
= (x + y)15
d) (w3z4)(w3z7) = w3z4w3z7
= w3w3z4z7
= w6z11
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-5
The Quotient Rule
For any nonzero number a and any positive
integers m and n for which m > n,
m
a
mn
a .
n
a
(To divide powers with the same base,
subtract the exponent of the denominator from
the exponent of the numerator.)
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
Example
Divide and simplify each of the following. (Here “simplify”
means express the product as one base to a power
whenever possible.)
14
9
7
7 9
(6
y
)
x
8
6
r
t
a)
b)
c)
d)
x3
83
(6 y ) 6
4r 3 t
Solution
9
x
a)
 x 9 3
x3
 x6
b)
87
7 3

8
83
 84
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1-7
Example
14
(6
y
)
14  6
8
c)
(6
y
)

(6
y
)

(6 y )6
7 9
7
9
6
r
t
6
r
t
d)
  3
3
4r t 4 r t
6 7 3 91 3 4 8
  r t  r t
4
2
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-8
The Exponent Zero
For any real number a, with a ≠ 0,
a  1.
0
(Any nonzero number raised to the 0
power is 1.)
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
Example
Simplify: a) 12450
d) (1)80 e) 90.
b) (3)0
c) (4w)0
Solution
a) 12450 = 1
b) (3)0 = 1
c) (4w)0 = 1, for any w  0.
d) (1)80 = (1)1 = 1
e) 90 is read “the opposite of 90” and is
equivalent to (1)90: 90 = (1)90 = (1)1 = 1
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-10
The Power Rule
For any number a and any whole numbers m
and n,
(am)n = amn.
(To raise a power to a power, multiply the
exponents and leave the base unchanged.)
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
Example
Simplify: a)(x3)4
b) (42)8
Solution
a) (x3)4 = x34
= x12
b) (42)8 = 428
= 416
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-12
Raising a Product to a Power
For any numbers a and b and any whole
number n,
(ab)n = anbn.
(To raise a product to a power, raise each
factor to that power.)
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
Example
Simplify: a)(3x)4
b) (2x3)2
c) (a2b3)7(a4b5)
Solution
a) (3x)4 = 34x4
= 81x4
b) (2x3)2 = (2)2(x3)2
= 4x6
c) (a2b3)7(a4b5) = (a2)7(b3)7a4b5
= a14b21a4b5
= a18b26
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Multiplying exponents
Adding exponents
1-14
Raising a Quotient to a Power
For any real numbers a and b, b ≠ 0, and
any whole number n,
n
n
a
a
 
   n.
b
b
(To raise a quotient to a power, raise the
numerator to the power and divide by the
denominator to the power.)
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
Example
Simplify: a)  w 
3
 
4
b)  3 
c)  2a 
4
5
 5
b 
2
4
b


Solution
3
3
w3
w w
a)  4   43  64
 2a 
c)  4 
 b 
4
34
 3
b)  5   5 4
 b  (b )

5
2
(2a5 )2
 4 2
(b )
22 (a5 )2 4a10

 8
4 2
b
b
81
81

b54 b 20
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.
1-16
Definitions and Properties of Exponents
For any whole numbers m and n,
1 as an exponent:
0 as an exponent:
The Product Rule:
a1 = a
a0 = 1
The Quotient Rule:
am
 a mn
n
a
(am)n = amn
The Power Rule:
Raising a product to a power:
Raising a quotient to a power:
a m  a n  a mn
(ab)n = anbn
n
an
a
   n
b
b
Copyright © 2014, 2010, and 2006 Pearson Education, Inc.