Laws of Exponents: Dividing Monomials
Division Rules for
Exponents
Laws of Exponents: Dividing Monomials
Division Rules of Exponents
Essential Questions
How do I divide powers with the
same bases?
How do I simplify expressions with
negative and zero exponents?
Laws of Exponents: Dividing Monomials
Rules and Properties
Quotient-of-Powers Property
For all nonzero real numbers x and all
integers m and n, where m > n,
xm
m–n
=
x
1.
xn
When dividing like bases,
subtract the exponents.
5
x
Examples:
5–3
x2
=
x
=
x3
Laws of Exponents: Dividing Monomials
Examples
Use the properties of exponents to simplify
expressions containing fractions.
7y3
x
2.
= x6y
xy 2
3
5
2x
4x
3.
6x2 = 3
Subtract the exponents
for the x (7 -1= 6)
Subtract the exponents
for the y (3 -2 = 1)
Reduce the coefficients.
Subtract the exponent of
the variables.
Laws of Exponents: Dividing Monomials
Do These Together
4.
x6
2
=
x
x4
5.
x3y7
= x2y3
xy4
6.
5x7y3z6
7.
15x3yz4
10x3y4
6xy4
x4y2z2
=
3
=
5x2
3
Laws of Exponents: Dividing Monomials
TRY THESE
8.
x8
5
=
x
x3
9.
x4y7
x4y2
= y5
2y3z3
4y6z8
3x
6x
10.
=
2x2y3z5
11.
18x5y9
12x3y3
=
3x2y6
2
Laws of Exponents: Dividing Monomials
By applying the
product of powers
property to the
following example, we
find that:
3 3 3
0
7
3
3 3
0
37
7

3
7
37
30  1
0 7
Zero Property of
Exponents
7
A nonzero number to the
zero power is 1:
We can then divide both
sides of the equation by
37 to determine the value
of 30
a  1, a  0
0
Laws of Exponents: Dividing Monomials
Evaluate the following expressions.
A.
 7
0
B. 4  4
2
2
C. 2  5
0
3
 3
D.  
 8
0
E. 00
Solutions
A. 1
B. 4  4
2
2
4
0
C. 20  5 3  1  125
1
D. 1
E. 0
0
is undefined.
 125
Laws of Exponents: Dividing Monomials
an  an  1
By applying the product of
powers property to the
following example, we find
that:
a a
n
n
a
n  (  n)
a
1
0
We can then divide both
sides of the equation by an to
determine the value of a-n
an  an
a

n
an 
a
-n
1
an
1
a
n
is the reciprocal
n
of a :
a
n

1
a
n
,
a0
Laws of Exponents: Dividing Monomials
Evaluate the following
expressions.
2
A. 3
B.
1
4
-3
Rewrite the following
expressions using positive
exponents.
A. 5 x 4
Solutions
A. 32 
B.
1
4
32
1

9
3
4
B.
A. 5 x
1
a -3
4
b -5
 5

3
 64
B.
a 3
b 5


1
a3
b5
a3
5
x4
 b5
1
x4
Laws of Exponents: Dividing Monomials
1) Evaluate the following expressions.
 3
A. -  
 5
0
B.
3
2
3
C. 9
4
9
7
 
D. 4
2 2
E.
1
3
0

12
2
2) Rewrite the following expressions with
positive exponents.
A. y
6
B. 7c
4
C.
2s 3
r
2
D. ( 5a )
3
 
E. 3x
1 4
Laws of Exponents: Dividing Monomials
0
 3
A.      1
 5
B.
3
2
3
 3 2
D.
C. 9
4
 256
3
9 9
7
4  7
9
 243
3
 4 2 2
4
 3 8
 24
4
 
4
2 2
E.
1
3
2
 12  3  1
0
2
 9
Laws of Exponents: Dividing Monomials
A. y
6
B. 7c

4
C.
r
2
y
D.
6
c4
7
c
 5 a
3
4
 2s 3  r 2
 2s r
3 2


1
 7

2s 3
1
E.

3x

1 4
1
 5 a
3
1
125 a 3
 34  x 14

1
 x4
34
x4

81