Slide 1- 1Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Properties of Exponents
The Product and Quotient Rules
The Zero Exponent
Negative Integers as Exponents
Raising Powers to Powers
Raising a Product or Quotient to a Power
1.6
Slide 1- 3Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplying with Like Bases: The Product Rule For any number a and any positive integers m and n,
(When multiplying powers, if the bases are the same, keep the base and add the exponents.)
m n m na a a
Slide 1- 4Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
8 3 3 7 2 11(a) ; (b) ( 2 )(7 ) z z x y x y
Solution
Multiply and simplify:
8 3 8 3 11(a) z z z z 3 7 2 11 3 2 7 11(b) ( 2 )(7 ) ( 2) 7 x y x y x x y y
3 2 7 1114 x y
5 1814 x y
Slide 1- 5Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Dividing with Like Bases: The Quotient Rule For any nonzero number a and any positive integers m and n, m > n,
(When dividing powers, if the bases are the same, keep the base and subtract the exponents.)
mm n
n
aa
a
Slide 1- 6Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
15 6 8
7 2 5
18(a) ; (b)
9
m x y
m x y
Solution
Divide and simplify:
1515 7 8
7
(a)
mm m
m
6 86 2 8 5
2 5
18(b) 2
9
x yx y
x y
4 32x y
Slide 1- 7Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Zero Exponent For any nonzero real number a,
(Any nonzero number raised to the zero power is 1. 00 is undefined.)
0 1.a
Slide 1- 8Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
0 0 0(a) ; (b) 2 ; (c) ( 2 ) . y y y
Solution
Evaluate each of the following for y = 5:
0 0(a) 5 1y
0 0(b) 2 2 5 2y
0 0 0(c) ( 2 ) ( 2 5) ( 10) 1y
Slide 1- 9Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Negative Exponents For any real number a that is nonzero and any integer n,
(The numbers a-n and an are reciprocals of each other.)
1nn
aa
Slide 1- 10Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
2 6 3
4
1(a) 12 (b) 2 (c) x y
m
Express using positive exponents and simplify if possible.
2
2
1 1(a) 12
12 144
36 3 3
6 6
1 2(b) 2 2
yx y y
x x
( 4) 4
4
1(c) m m
m
Solution
Slide 1- 11Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Factors and Negative Exponents
For any nonzero real numbers a and b and any integers m and n,
(A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed.)
.n m
m n
a b
b a
Slide 1- 12Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
1 6 3
4
5 .
x y
w
Write an equivalent expression without negative exponents:
Solution1 6 3 4 3
4 6
5 =
5
x y w y
w x
Slide 1- 13Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
22 13
4(a) ; (b) .
mx x
m
The product and quotient rules apply for all integer exponents.
22 ( 4) 2
4(b)
mm m
m
Solution
Simplify:
2 13 2 ( 13) 11
11
1(a) x x x x
x
Slide 1- 14Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Raising a Power to a Power: The Power Rule For any real number a and any integers m and n,
(To raise a power to a power, multiply the exponents.)
( ) .m n mna a
Slide 1- 15Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
2 4 3 8 5 16(a) ( ) ; (b) (5 ) ; (c) ( ) . x m
Simplify:
2 4 2 4 8
8
1(a) ( )x x x
x
3 8 3 8 24(b) (5 ) 5 5
5 16 5 ( 16) 80(c) ( ) m m m
Solution
Slide 1- 16Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Raising a Product to a Power
For any integer n, and any real numbers a and b for which (ab)n exists,
(To raise a product to a power, raise each factor to that power.)
( ) .n n nab a b
Slide 1- 17Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
4 3 5 3(a) (3 ) ; (b) (5 ) . y x y
Simplify:
Solution4 4 4 4(a) (3 ) 3 81y y y
3 5 3 3 3 3 5 3(b) (5 ) 5 ( ) ( ) x y x y 9 15125 x y
15
9
125
y
x
Slide 1- 18Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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,
(To raise a quotient to a power, raise both the numerator and denominator to that power.)
.n n
n
a a
b b
Slide 1- 19Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
2 53
4 2
9(a) ; (b) .
xy
y z
Simplify:
Solution2 2
4 4 2 8
9 9 81(a)
( )y y y
53 3 5 5 15 15
2 2 5 10 5 10
( )(b)
( )
xy xy x y y
z z z x z
