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”

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
( )
( )
mnn
m
nm
n
m
m
mm
mnnm
mmmnmnm
xx
x
thenmnifb
x
x
x
thennmifa
y
x
y
x
xx
yxxyxxx
−
−
+
=>
=>
=





=
==⋅
1
,.5
,.5
.4.3
.2.1

3. Other Properties of Exponents
10
=x
Any single number or variable is always to
the first power
( )11111
22233 xxxaa ====
x
x
11
=−

4. Basic Examples
=⋅ 32
xx =+32
x 5
x
( ) =
34
x =⋅34
x 12
x
( ) =
3
xy
33
yx

5. =





3
y
x
3
3
y
x
=4
7
x
x
=
−
1
47
x 3
x
=7
5
x
x
=−57
1
x 2
1
x
Basic Examples

6. More Examples
=⋅ 43
72 aa ( ) =⋅ +43
72 a 7
14a
=⋅⋅− 232
285 rrr ( ) =⋅⋅− ++ 232
285 r 7
80r−
( ) =
3
3xy =333
3 yx 33
27 yx
=





2
3
2
b
a
=22
22
3
2
b
a
2
2
9
4
b
a
( ) =
352
2 nm =⋅⋅⋅ 353231
2 nm =1563
2 nm 156
8 nm
=
x
x
2
8 4
=
−
12
8 14
x 3
4x
=5
3
3
9
z
z
=−35
1
3
9
z
=2
1
3
x 2
3
x

7. More Examples
=⋅ 223
73 xyzzyx ( ) =⋅ +++ 211213
73 zyx 334
21 zyx
( ) =−⋅⋅− 32
238 xyxyxy ( ) =−⋅⋅− ++++ 312111
238 yx 63
48 yx
( ) ( ) =
22232
23 xyyx ( )( ) =⋅⋅⋅⋅⋅⋅ 222121232221
23 yxyx
=⋅ 4264
49 yxyx ( ) =⋅ ++ 4624
49 yx 106
36 yx
=





3
2
3
3
5
ab
ba
=⋅⋅⋅
⋅⋅⋅
323131
313331
3
5
ba
ba
=633
393
3
5
ba
ba
=63
39
27
125
ba
ba
=−
−
36
39
27
125
b
a
3
6
27
125
b
a

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