The document discusses the laws of exponents, which define the mathematical operations that can be performed on variables with the same base but different exponents. It provides several examples of applying the laws of exponents, such as combining like terms, distributing exponents across multiplication, and negative exponents becoming fractions. It also covers some basic properties of exponents, such as any single number or variable being to the first power and anything to the zero power equaling one.

1. Laws of Exponents
Whenever we have variables which
contain exponents and have equal bases,
we can do certain mathematical
operations to them. Those operations
are called the “Laws of Exponents”
bx
b = base x = exponent

2. Laws of Exponents
 
 
m
n
n
m
n
m
n
m
m
m
m
mn
n
m
m
m
m
n
m
n
m
x
x
x
then
m
n
if
b
x
x
x
then
n
m
if
a
y
x
y
x
x
x
y
x
xy
x
x
x




















1
,
.
5
,
.
5
.
4
.
3
.
2
.
1

3. Other Properties of Exponents
1
0

x
Any single number or variable is always to
the first power
 1
1
1
1
1
2
2
2
3
3 x
x
x
a
a 



x
x
1
1



4. Basic Examples

 3
2
x
x 
3
2
x 5
x
  
3
4
x 
3
4
x 12
x
  
3
xy
3
3
y
x

5. 








3
y
x
3
3
y
x

4
7
x
x


1
4
7
x 3
x

7
5
x
x

5
7
1
x 2
1
x
Basic Examples

6. More Examples

 4
3
7
2 a
a   
 4
3
7
2 a 7
14a



 2
3
2
2
8
5 r
r
r   


 
 2
3
2
2
8
5 r 7
80r

  
3
3xy 
3
3
3
3 y
x 3
3
27 y
x







2
3
2
b
a

2
2
2
2
3
2
b
a
2
2
9
4
b
a
  
3
5
2
2 n
m 


 3
5
3
2
3
1
2 n
m 
15
6
3
2 n
m 15
6
8 n
m

x
x
2
8 4


1
2
8 1
4
x 3
4x

5
3
3
9
z
z

3
5
1
3
9
z

2
1
3
x 2
3
x

7. More Examples

 2
2
3
7
3 xyz
z
y
x   
 

 2
1
1
2
1
3
7
3 z
y
x 3
3
4
21 z
y
x
 



 3
2
2
3
8 xy
xy
xy   



 


 3
1
2
1
1
1
2
3
8 y
x 6
3
48 y
x
    
2
2
2
3
2
2
3 xy
y
x   





 2
2
2
1
2
1
2
3
2
2
2
1
2
3 y
x
y
x

 4
2
6
4
4
9 y
x
y
x   
 
 4
6
2
4
4
9 y
x 10
6
36 y
x









3
2
3
3
5
ab
b
a







3
2
3
1
3
1
3
1
3
3
3
1
3
5
b
a
b
a

6
3
3
3
9
3
3
5
b
a
b
a

6
3
3
9
27
125
b
a
b
a



3
6
3
9
27
125
b
a
3
6
27
125
b
a