dividing monomials honors math – grade 8. quotient of powers look for a pattern in the exponents....
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Dividing Monomials
Honors Math – Grade 8
Quotient of PowersLook for a pattern in the exponents.
3 factors
5 factors
KEY CONCEPT Quotient of Powers
To divide two powers with the same base, subtract their exponents.
87157
15
bbb
b
3
5
4
4
444
44444
44 24
Definition of Power
Cancel common factors.
What is the pattern?
Simplify3
85
ab
ba
))(( 3815 ba54ba
Group powers that have the same base.
Quotient of Powers
Simplify
3
85
ab
ba
3
85
b
b
a
a
A variable with no exponent indicated can be written as a power of 1.
Simplify36
127
yx
yx
))(( 31267 yx9xy
Group powers that have the same base.
Quotient of Powers
Simplify
36
127
yx
yx
3
12
6
7
y
y
x
x
Power of a Quotient
Look for a pattern in the exponents.
3 factors
KEY CONCEPT Power of a Quotient
To find a power of a quotient, find the power of the numerator and the power of the denominator.
m
mm
b
a
b
a
3
5
2
5
2
5
2
5
2
555
222
3
3
5
2
Definition of Power
Multiply
What is the pattern?
3 factors
Simplify42
3
2
p
Power of a Quotient
Power of a Product
Power of a Power
4
42
3
2p
42
3
2
p
4
424
3
2 p
81
16 8p
Simplify25
6
5
yx
Power of a Quotient
Power of a Product
Power of a Power
2
25
6
)5( yx
2
2252
6
)()5( yx
36
25 210 yx
25
6
5
yx
The Zero Exponent
Method 2Method 1
KEY CONCEPT Zero Exponent
Any number raised to the zero power is equal to 1.
1)25.0(1 00 andb
4
4
2
2 Definition of Power
Cancel common factors.
Simplify
0444
4
222
2
12222
2222
2
24
4
Quotient of Powers
Simplify0
7
5
8
3
xy
yx
Definition of Zero Exponent.
Recall that anything to the zero power is equal to 1
10
7
5
8
3
xy
yx
Even an entire expression raised to the zero power equals 1!
Simplify2
40
y
yx
Zero Exponent
Quotient of Powers
2
4 )(1
y
y
2
4
)1(y
y
24y
2
40
y
yx
Group powers that have the same base.
2y
Negative Exponents
Method 2Method 1
KEY CONCEPT Negative Exponent
For any nonzero number a and any integer n, a-n is the reciprocal of an. In addition, the reciprocal of a-n is an.
332
2 1
25
1
5
15 m
mand
5
2
8
8 Definition of Power
Cancel common factors.
Simplify
3525
2
888
8 35
2
8
1
88888
88
8
8
Quotient of Powers
Simplify5
23
d
cb
Write as a product of fractions.
Negative Exponents Definition
Multiply Fractions
5
23 1
11 d
cb
11
1 52
3
dc
b
3
52
b
dc
5
23
d
cb
Simplify572
74
21
3
cba
ba
Write as a product of fractions.
Quotient of Powers and Negative
Exponent Definition
Simplify
57
7
2
4 1
21
3
cb
b
a
a
))()((7
1 57724 cba
506
7
1cba
572
74
21
3
cba
ba
56
)1(1
7
1c
a
6
5
7
1
a
c
Negative Exponent and Zero Exponent
DefinitionsMultiply
Simplify53
42
12
3
sqr
rsq
Write as a product of fractions.
Quotient of Powers
Simplify
5
4
3
2
12
3
s
s
r
r
q
q
))()((4
1 543112 srq
943
4
1srq
943
)(1
4
1sr
q
3
94
4q
srNegative Exponent
DefinitionMultiply
53
42
12
3
sqr
rsq