Chapter 8
Coursepack:
Exponential and Logarithmic
Functions
Honors Algebra 2
C. Bianchi, 2019
388
Preparation for Chapter 8: Review of Rational
Exponents
PROPERTIES OF EXPONENTS
x
a
•xb
= xa + b Multiplyin
...
Chapter 8
Coursepack:
Exponential and Logarithmic
Functions
Honors Algebra 2
C. Bianchi, 2019
388
Preparation for Chapter 8: Review of Rational
Exponents
PROPERTIES OF EXPONENTS
x
a
•xb
= xa + b Multiplying powers: Add exponents
b
a
x
x
= x
a – b
Dividing powers: Subtract exponents
(x
a
)
b
= x
ab Power of a power: Multiply exponents
(xy)
a = x
a
y
a Power of a product: Distribute the exponent
a
y
x
=
b
a
x
x
Power of a quotient: Distribute the exponent
x
-a
=
a
x
1
Negative exponent: Take the reciprocal
x
0
= 1 (x ≠ 0)
EXAMPLE 1: 7
-2
4 2 3 ( ) xy z
− 2 3 5 4 ( )( ) x y x y
=
2
1
7
=
3 4 3 2 3 ( )( ) ( ) x y z
−
=
2 3 5 4 4 ( )( ) ( ) x y x y
=
1
49
=
3 12 6
x y z
−
=
2 3 20 4
x y x y •
=
3 12
6
x y
z
=
22 7
x y
Recall that a rational (fractional) exponent represents some type of root (square root, cube root,
etc.).
EXAMPLE 2:
1
2 81
1
3
( 8) −
1
4 16
=
81
=
3 −8
=
4
16
= 9 = –2 = 2
DEFINITION OF RATIONAL EXPONENT:
1
n n
x x = ( )
m
m
n n
x x =
The second definition indicates that the numerator represents a power, while the denominator
represents a root. For example,
2
3 27
means taking the second power (square) of a third (cube)
root.
389
EXAMPLE 3:
2
3 27
5
4 16
−
=
( )
2
3
27
=
5
4
1
16
= (3)2 =
( )
5
4
1
16
= 9 =
5
1
2
=
1
32
Simplify each expression as much as possible. There should be no negative exponents in any
final answer. Do not use a calculator.
1.
2
13−
2.
2
15−
3.
3
5
−
4.
3
4
−
5.
3
6
7
−
6.
4
5
3
−
7.
3 8
x x •
8.
6 14
x x
−
•
9.
7 6 ( ) x
−
10.
8 9 ( ) x
11.
4 6 3 2 ( ) x y z
−
12.
9 6 2 4 ( ) x y z
− −
13.
6 3 8 4 ( )( ) x y x y
−
14.
3 2 6 5 ( )( ) x y x y
−
15.
15 3
8 10 3
x y z
x y z
−
16.
6 7
13 5 6
x y z
x y z
−
17.
1
2 196
18.
1
2 289
−
19.
1
3 216
−
20.
1
3 343
21.
3
4 256
22.
4
3 125
23.
2
3 27
−
24.
3
2 16
−
25.
1 1
3 3 4(5) 9(5) +
26.
1 1
4 4 6(7) 8(7) −
27.
1
6 2
(72 ) x
28.
1
10 2
(68 ) x
29.
1
6 3
( 40 ) − x
30.
1
3 3
( 56 ) − x
31.
6
4
( 7)
( 7)
x
x
−
−
32.
12
5
( 3)
( 3)
x
x
+
+
33. State the domain restriction for Number 31.
34. State the domain restriction for Number 32.
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