Recall
A xA is called the
_____________x is called the _____________
The operation involved is ___________
6.1: Using Properties of Exponents• Properties of Exponents:• 1. Product of powers• 2. Power of a power • 3. power of a product• 4. negative exponent• 5. zero exponent• 6. quotient of powers• 7. power of a quotient
nmnm aaa • When you multiply same bases,
you add exponents!!!!
2 7
42
3
3 2 4
1.) 4 4
2.)
3.) (6 )
4.) ( )
x
a
w v w
2 74 94 262,1442 4x 8x
1 3 1 36 a
3 8 4( )w v w 3 4 8w v 7 8w v
3 2 4 1 4( )w v w
3 36 a 3216a
Zero exponent:
Evaluate:01.) 5 1
02.) 0 !!!undefined
03.) ( 2) 1
014.) ( ) 1
9
ANYTHINGTO THE ZERO POWERIS ONE!!!!!!
Evaluating Exponential Expressions:
4 41. 6 6 4 46 06 1 232. 2
3 22 62 64
23. 3 2
2( 6) 2
1
6
1
36
2 3
1 2
2
YOU TRY: 1. 4 4
2. (3 )
3. (2 5)
SIMPLIFYING EXPONENTIALEXPRESSIONS:
***Basically, when we simplify, we make sure that our final answer has NO negative exponents in it!
2 31. 2x y What are my negative exponents???Yes, the x and the y bases so those are theONLY one’s that I will move to the denominator!
2
32.
c
d
So, both are negative so the c goes down to the denominator and the d goes up to the numerator!3
2
d
c
2 3
2
x y
Simplify each expression completely.
853
92
34
336
.4
48 .3
45 .2
.1
xxx
aab
xx
yx
918 yx7000,40 x
211256 ba16x
Simplify the expression, only using positiveexponents:
3 4
3 7
3
2 4
0
2
2 4
3
4
1.
2. ( )
3. (2 )
4. ( )
5. 9
16.
5
7. 8
98.
x x
a
d
mn n
x y
p
q
Division Properties of Exponents
** *Quotient of Powers:To divide powers that have the same base, you subtract the exponents.This is called the quotient of powers property.
EXAMPLES:5
4
61.
65 46 6
4
6
82.
84 68 28 2
1
8 1
64
3
53.
y
y3 5y 2y 2
1
y
3
2
( 3)4.
3
3 23
13 3
4
45.
x
x4 4x 0x 1YOU TRY!!!
4 11 8 8
1 18 3 8
5 ( 2)1. 2. 3. 4.
5 ( 2)
m s
m s
***Power of a Quotient: Remember, that when you have a base to a power raised to a power, that youmultiply exponents!
EXAMPLES:22
1. ( )3
2
2
2
3
4
9
Remember, that you only SUBTRACT exponents,the bases HAVE TO BE THE SAME!!!! SO, that’s why we just multiply 2 by itself and 3 byitself
33
2.y
3
3
( 3)
y
3
27
y
37
3.4
3
3
7
4
3
3
4
7
64
343
Simplify Expressions using MultipleProperties:
2 2
4
2 91.
3
x y xy
x y
3 3
4
18
3
x y
xy 3 1 3 46x y 2 16x y
26x
y
4
2
22.
x
y
4
2 4
(2 )
( )
x
y
4 4
2 4
2 x
y 4
8
16x
y
YOU TRY:34
3 3 3
3 51. 2.
xy y x
x xy y
Simplify Expressions with NegativeExponents:
32
11.
x x
y y
2 3
1 1 3
x x
y y
6
1 3
x x
y y
1 ( 6)
1 ( 3)
x
y
5
4
x
y
4
5
y
x
12 4
22.
y x
x y
2 4 1
2 1 1
y x
x y
2 4
2 1
y x
x y
1
2 2 4
1 y
x y x
6 2
y
x y
1 2
6
y
x
1
6
y
x
6
1
x y
Simplify each expression completely.
?2or 2 larger, is Which .5
22 .4
.3
32 .2
3 .1
22
25342
47
1220
42
--
cbbc
y y
yx
xy
886561 yx
3
22 yx
11
1
y
181064 cb 22