Exponents
5
exponent
3
Exponential
Form
base
Example: 125  53 means that 53 is the exponential
form of the number 125.
53 means the product of 3 factors of 5 = 5 x 5 x 5
The Laws Of Exponents
The laws of exponents tell
us how to simplify
expressions involving
exponents.
There are 7 laws of exponents.
Exponential form: raising a base to a power stands for
repeated multiplication of the base times itself.
x  x  x  x  x  x  x  x
n
n times
n factors of x
Example: 5  5  5  5
3
Exponential form
Standard form
Questions
What is a power?
What is a base?
What is exponential form?
What is standard form?
What are the laws of exponents?
How many laws of exponents are there?
Write in standard form.
Write in exponential
form.
Questions
Give an example of a power.
Give an example of a base.
Give an example of exponential form.
Give an example of standard form.
The Laws of Exponents:
#1: Multiplying Bases:
If you are multiplying bases with
the same base, KEEP the BASE & ADD the EXPONENTS!
x x  x
m
So, I get it!
When you
multiply
bases, you
add the
exponents!
n
mn
2 6  23  2 6  3  29
 512
Try these:
1. 3  3 
2
2
2. 52  54 
3.
a a 
5
2
4. 2s  4s 
2
7
5. (3)  (3) 
2
6.
3
s t s t 
2 4
7 3
SOLUTIONS
2
2 2
4
a a  a
5 2
a
1. 3  3  3  3  81
2 4
6
2
4
2. 5  5  5  5
2
3.
5
2
4. 2s  4s  2  4  s
2
7
5. (3)  (3)  (3)
2
6.
3
s t s t 
2 4
7 3
s
7
27
23
 8s
 (3)  243
27 43
t
9
5
s t
9 7
Questions
What is the first law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.
#2: Dividing Bases: When dividing bases with the same
base, KEEP the BASE & SUBTRACT the EXPONENTS!
m
x
m
n
mn

x

x

x
n
x
So, I get it!
When you
divide bases,
you subtract
the
exponents!
6
2
6 2
4

2

2
2
2
 16
Try these:
12
7.
s

4
s
9
8.
3

5
3
12 8
9.
s t

4 4
st
5 8
10.
36a b

4 5
4a b
SOLUTIONS
12
7.
8.
9.
10.
s
12 4
8
s

s

4
s
9
3
9 5
4
3

3
 81

5
3
12 8
s t
12 4 8 4
8 4
s t s t

4 4
st
5 8
36a b
5 4 85
3
36

4

a
b

9
ab

4 5
4a b
Questions
What is the second law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.
#3: Power of a Power: If you are raising a power to a
power, you multiply the exponents!
x 
n
m
So, when I
take a Power
to a power, I
multiply the
exponents
x
mn
32
(5 )  5
3
2
5
6
Try these:
 

 

2 5
1. 3
2. a
3 4
SOLUTIONS
 
2 5
1. 3
 
2. a
3 4


10
3
a 12
Questions
What is the third law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.
#4: Product Rule of Exponents: Raising a product to
a power means raising all of the factors to the power and then
multiplying.
 xy 
So, the power of
the product is
the product of
the powers.
n
x y
n
n
(ab)  a b
2
2
2
Try these:
 
3. 2a

2
2 3


5 3 2
4. 2 a b

5. (3a ) 
2 2
 
2 4 3
6. s t

SOLUTIONS
 
3. 2a

2
2 3
3
 2 a

5 3 2
4. 2 a b
23
 8a
22 52 32
4 10 6
10 6
 2 a b  2 a b  16a b
5. (3a )   3  a
2
2 2
 
2 4 3
6. s t
6
23 43
s t
22
s t
 9a
6 12
4
Questions
What is the fourth law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.
#5: Quotient Rule of Exponents: Raising a quotient to a
power means raising the numerator and denominator to the power.
n
 x
x
   n
y
 y
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
n
4
16
2 2
   4 
81
3 3
4
Try these:
5
s
7.   
t
2
3 
8.  5  
3 
8 2
 st 
9.  4  
 rt 
9
 36a b
10. 
4 5
 4a b
5 8
2

 

SOLUTIONS
5
s
7.   
t
5
s
5
t
2
3 
8.  5   34
3 
9
 
2
3
8
2
4 2
2 8
 st 


st
s
t
9.  4   
  2

r
 rt 
 r 
8
 36a b
10 
4 5
 4a b
5 8
2

  9ab3



2
2 32
9 a b
2
 81a b
2 6
Questions
What is the fifth law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.
#6: Law of Negative Exponents: A base raised to a
negative power is the reciprocal of the base to the positive power.
So, when I have a
negative exponent, I
take the reciprocal
and change the power
to a positive.
x
m
1
 m
x
1
1
5  3 
5
125
and
3
1
2

3
9
2
3
Try these:
2.
2
y y
4
3.
a 

2
7
5 1

4. s  4s 

2
5. 3x y

3 4

1
2 
7.   
 x 2
 39 
8.  5  
3 
2
2
s t 
9.  4 4  
s t 
2
5
 36a 
10.  4 5  
 4a b 
2 2
SOLUTIONS
 
1
3. a
 5
a
5
2
7
4. s  4s  4s
5 1

2
5. 3x y

3 4

4
 3 x y
8
12

8
x

81y12
SOLUTIONS
1
1
x
2 
4
7.      
x
4


x
 
9 2
1
3 
4 2
8
8.  5   3   3  8
3
3 
2
2


s t 
 2  2 2
4 4
s t
9.  4 4   s t
s t 
10
2
5
b

2

2
10
 36a 
9
a
b

2


10.  4 5  
81a
4
a
b


2 2
Questions
What is the sixth law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.
#7: Zero as an exponent: Anything to the zero power is 1.
x 1
0
So whenever
I raise
anything to
the zero
power it is
always 1!
50  1
and
a0  1
and
(5 a ) 0  1
Try these:
1.
2a b
6.
s t 
2
2 4 0
0


SOLUTIONS


0
1. 2a b  1
2
 
2 4 0
6. s t
1
Questions
What is the seventh law of exponents?
When can I use it?
When can I not use it?
Give an example of where I use it.
Give an example of where I can’t use it.
Why does it work? Give an example.