1. ExponentsExponents
{ 3
5power
base
exponent
3 3
means that is the exponential
form of t
Example:
he number
125 5 5
.125
=

2. The Laws of Exponents:The Laws of Exponents:
#1: Exponential form: The exponent of a power indicates how
many times the base multiply itself.
n
n times
x x x x x x x x
−
= × × ×××× × × ×144424443
3
Example: 5 5 5 5= × ×

3. The Laws of Exponents:The Laws of Exponents:
#2: Multiplicative Law of Exponents: If the bases are the same
And if the operations between the bases is multiplication, then the
result is the base powered by the sum of individual exponents .
m n m n
x x x +
× =
3 4 3 4 7
Example: 2 2 2 2+
× = =
( ) ( )3 4
7
Proof: 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
× = × × × × × × =
× × × × × × =

4. The Laws of Exponents:The Laws of Exponents:
#3: Division Law of Exponents: If the bases are the same
And if the operations between the bases is division, then the
result is the base powered by the difference of individual
exponents .
m
m n m n
n
x
x x x
x
−
= ÷ =
4
4 3 4 3 1
3
5
Example: 5 5 5 5 5
5
−
= ÷ = = =
4
3
5 5 5 5 5
Proof: 5
5 5 5 5
× × ×/
= =
× ×/

5. The Laws of Exponents:The Laws of Exponents:
#4: Exponential Law of Exponents: If the exponential form
is powered by another exponent, then the result is the base
powered by the product of individual exponents.
( )
nm mn
x x=
( )
23 3 2 6
Example: 4 4 4×
= =
( ) ( ) ( ) ( )
2 23
6
Proof: 4 4 4 4 4 4 4 4 4 4
4 4 4 4 4 4 4
= × × = × × × × × =
= × × × × × =

6. The Laws of Exponents:The Laws of Exponents:
#5: Product Law of Exponents: If the product of the bases
is powered by the same exponent, then the result is a multiplication
of individual factors of the product, each powered by the given
exponent.
( )
n n n
xy x y= ×
( )
22 2
2 2
2
Proof: 2 3 4 9
Example: 36 6 2 3 2 3
36
= = × = ×
× = × =

7. The Laws of Exponents:The Laws of Exponents:
#6: Quotient Law of Exponents: If the quotient of the bases
is powered by the same exponent, then the result is both
numerator and denominator , each powered by the given
exponent.
n n
n
x x
y y
 
= ÷
 
3 3
3
2 2
Example:
7 7
 
= ÷
 

8. The Laws of Exponents:The Laws of Exponents:
#7: Negative Law of Exponents: If the base is powered by the
negative exponent, then the base becomes reciprocal with the
positive exponent.
1m
m
x
x
−
=
3
3
1 1
Example #1: 2
2 8
−
= =
3
3
3
1 5
Example #2: 5 125
5 1−
= = =

9. The Laws of Exponents:The Laws of Exponents:
#8: Zero Law of Exponents: Any base powered by zero exponent
equals one
0
1x =
( )
0
0
0
Example: 112 1
5
1
7
1flower
=
 
= ÷
 
=

10. The Laws of Exponents:The Laws of Exponents:
#8: Zero Law of Exponents: Any base powered by zero exponent
equals one
0
1x =
( )
0
0
0
Example: 112 1
5
1
7
1flower
=
 
= ÷
 
=