1. Law of
Exponents
Donna G. Bautista
BSEd 4-C

2. ExponentsExponents
{ 3
5Power
base
exponent
3 3
means that is the exponential
form of t
Example:
he number
125 5 5
.125
=
53
means 3 factors of 5 or 5 x 5 x 5

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

4. #2: Multiplying Powers: If you are multiplying Powers
with the same base, KEEP the BASE & ADD the EXPONENTS!
m n m n
x x x +
× =
So, I get it!
When you
multiply
Powers, you
add the
exponents!
512
2222 93636
=
==× +

5. #3: Dividing Powers: When dividing Powers with the
same base, KEEP the BASE & SUBTRACT the EXPONENTS!
m
m n m n
n
x
x x x
x
−
= ÷ =
So, I get it!
When you
divide
Powers, you
subtract the
exponents!
16
22
2
2 426
2
6
=
== −

6. Try these:
=× 22
33.1
=× 42
55.2
=× 25
.3 aa
=× 72
42.4 ss
=−×− 32
)3()3(.5
=× 3742
.6 tsts
=4
12
.7
s
s
=5
9
3
3
.8
=44
812
.9
ts
ts
=54
85
4
36
.10
ba
ba

7. =× 22
33.1
=× 42
55.2
=× 25
.3 aa
=× 72
42.4 ss
=−×− 32
)3()3(.5
=× 3742
.6 tsts
8133 422
==+
725
aa =+
972
842 ss =×× +
SOLUTIONS
642
55 =+
243)3()3( 532
−=−=− +
793472
tsts =++

8. =4
12
.7
s
s
=5
9
3
3
.8
=44
812
.9
ts
ts
=54
85
4
36
.10
ba
ba
SOLUTIONS
8412
ss =−
8133 459
==−
4848412
tsts =−−
35845
9436 abba =×÷ −−

9. #4: Power of a Power: If you are raising a Power to an
exponent, you multiply the exponents!
( )
nm mn
x x=
So, when I
take a Power
to a power, I
multiply the
exponents
52323
55)5( == ×

10. #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= ×
So, when I take
a Power of a
Product, I apply
the exponent to
all factors of
the product.
222
)( baab =

11. #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
 
= ÷
 
So, when I take a
Power of a
Quotient, I apply
the exponent to
all parts of the
quotient.
81
16
3
2
3
2
4
44
==






12. Try these:
( ) =
52
3.1
( ) =
43
.2 a
( ) =
32
2.3 a
( ) =
2352
2.4 ba
=− 22
)3(.5 a
( ) =
342
.6 ts
=





5
.7
t
s
=





2
5
9
3
3
.8
=





2
4
8
.9
rt
st
=





2
54
85
4
36
.10
ba
ba

13. ( ) =
52
3.1
( ) =
43
.2 a
( ) =
32
2.3 a
( ) =
2352
2.4 ba
=− 22
)3(.5 a
( ) =
342
.6 ts
SOLUTIONS
10
3
12
a
6323
82 aa =×
6106104232522
1622 bababa ==×××
( ) 4222
93 aa =×− ×
1263432
tsts =××

14. =





5
.7
t
s
=





2
5
9
3
3
.8
=





2
4
8
.9
rt
st
=





2
54
85
4
36
10
ba
ba
SOLUTIONS
( ) 62232223
8199 babaab == ×
2
8224
r
ts
r
st
=





( ) 824
33 =
5
5
t
s

15. #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
−
=So, when I have a
Negative Exponent, I
switch the base to its
reciprocal with a
Positive Exponent.
Ha Ha!
If the base with the
negative exponent is in
the denominator, it
moves to the
numerator to lose its
negative sign!
93
3
1
125
1
5
1
5
2
2
3
3
==
==
−
−
and

16. #8: Zero Law of Exponents: Any base powered by zero
exponent equals one.
0
1x =
1)5(
1
15
0
0
0
=
=
=
a
and
a
and
So zero
factors of a
base equals 1.
That makes
sense! Every
power has a
coefficient
of 1.

17. Try these:
( ) =
02
2.1 ba
=× −42
.2 yy
( ) =
−15
.3 a
=×− 72
4.4 ss
( ) =
−− 432
3.5 yx
( ) =
042
.6 ts
=





−12
2
.7
x
=





−2
5
9
3
3
.8
=





−2
44
22
.9
ts
ts
=





−2
54
5
4
36
.10
ba
a

18. SOLUTIONS
( ) =
02
2.1 ba
=× −42
.2 yy
( ) =
−15
.3 a
=×− 72
4.4 ss
( ) =
−− 432
3.5 yx
( ) =
042
.6 ts
1
2
2 1
y
y =−
5
1
a
5
4s
( ) 12
8
1284
81
3
y
x
yx =−−
1

19. =





−12
2
.7
x
=





−2
5
9
3
3
.8
=





−2
44
22
.9
ts
ts
=





−2
54
5
4
36
.10
ba
a
SOLUTIONS
4
4
1
x
x
=





−
( ) 8
824
3
1
33 == −−
( ) 44222
tsts =
−−−
2
10
1022
81
9
a
b
ba =−−