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. These laws include: (1) when multiplying the same base, add the exponents; (2) when dividing the same base, subtract the exponents; (3) when raising a base to a power, multiply the exponents. Examples are provided to demonstrate applying these laws to simplify exponential expressions.

10. 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

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

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

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

14. 
  


 

3
x
y
Basic Examples
3
3
x
y
7
 4
x
x
x 7 
4 
x
3 1
5
 7
x
x
1
 x 75
2
1
x

15. More Examples
 4 3 72aa    34 2 7 a 7 14a
    2 3 2 5r 8r 2r      232 5 8 2 r 7 80r 
   3 3 xy  3 3 3 3 x y 3 3 27y x

 


2
a
2
3
b
2 2
a
 2 2
2
3
b
2
2
4
9
b
a
  
2 5 3 2m n  13 23 53 2 m n  3 6 15 2 m n 6 15 8 n m

8 4
x
2
x
8 x 4 
1 
4x
3 2 1
3
 5
9
3
z
z
9
1
 3
z 53
2
1
3
z
3
z
 2

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

17. References
 http://www.wiziq.com/tutorial/8922-College-Algebra-Lecture-03
 www.worldofteaching.com
 http://www.intmath.com/Basic-algebra/3_Division-algebra.php