Transcript Exponent Rules - McCullough Junior High School


When a number, variable, or expression is
raised to a power, the number, variable, or
expression is called the base and the power is
called the exponent.
b
n


An exponent means that you multiply the
base by itself that many times.
For example
x4 =
x●x ●x●x
26 = 2 ● 2 ● 2 ● 2 ● 2 ● 2 = 64

When an expression does not have a visible
exponent its exponent is understood to be 1.
xx
1

When multiplying two expressions with the
same base you add their exponents.
b  b b
n

m
For example
nm
2 4
x x  x  x
2
1
2
1 2
3
2 2 2  2  2  2  8
2
4
6
b  b b
n

m
nm
Try it on your own:
3 7
1. h  h  h  h
2
2 1
3
2. 3 3  3  3
 3  3  3  27
3
7
10

When dividing two expressions with the same
base you subtract their exponents.
n

b
n

m

m
b
For example b
4
x
4 2
2

x

x
2
x
n

b
nm

b
m
b
Try it on your own:
6
h
6 2
4
3. 2  h  h
h
3
3
31
2
4.
 3 3  9
3

When raising a power to a power you multiply
the exponents
(b )  b
n m

nm
For example
(x )  x  x
2 2
22
4
(2 )  2  2  16
2 4
24
8
(b )  b
n m

nm
Try it on your own
3 2
5. (h )  h
32
22
h
6
6. (3 )  3  3  81
2 2
4

When using this rule the exponent can not be
brought in the parenthesis if there is addition
or subtraction
(x  2 )  x  2
2
2 2
4
You would have to use FOIL in these cases
4

When a product is raised to a power, each
piece is raised to the power
(ab)  a b
m

m m
For example
(xy)  x y
2
2
2
(2  5)  2  5  4 25  100
2
2
2
(ab)  a b
m

m m
Try it on your own
7. (hk )  h k
3
3 3
8. (2  3)  2  3  4 9  36
2
2
2

This rule is for products only. When using
this rule the exponent can not be brought in
the parenthesis if there is addition or
subtraction
( x  2)  x  2
2
2
You would have to use FOIL in these cases
2

When a quotient is raised to a power, both
the numerator and denominator are raised to
the power
m
m
a
a
   m
b
b

For example
3
x
x
   3
y
y
 
3
m

a
a
   m
b
b


Try it on your own
m
2
h
h
9.    2
k k
2
2
4
16
4
4
10.    2 
2
4
2
2

When anything, except 0, is raised to the zero
power it is 1.
a 1
0

For example
x 1
0
25 1
0
( if a ≠ 0)
( if x ≠ 0)
a 1
0

( if a ≠ 0)
Try it on your own
11. h  1
0
12. 1000  1
0
13. 0  0
0
( if h ≠ 0)


1
b

n
For example
b
1
2
x  2
x
1 1
2
3  2
3
9
If b ≠ 0, then
n


1
b

n
Try it on your own:
b
1
3
14. h  3
h
1 1
3
15. 2  3 
2
8
If b ≠ 0, then
n

The negative exponent basically flips the part
with the negative exponent to the other half
of the fraction.
 1  b 
2
 2      b
b   1 
2
 2   2x 
2
  2x
 2   
x   1 
2
 For
a problem to be
completely simplified there
should not be any negative
exponents
5
6d
2
5 9
4
1.

2
d
 2d  4
9
3d
d
2. 2e 4e  8e
4
5
45
 8e
9
 
3. q
4 5
q q
45
20
4. 2lp   2 l p  32l p
5
5 5
5
5
5
2
4
8
4
x y
( x y)
8 2 4  2
6 2

5.

x
y
x y
2
2 2
( xy)
x y
3 5 2
8 2
16
x
(x x ) (x )
169
7


6.

x
x
9
9
9
x
x
x
6 4 2
3 2
5 6
7. (m n ) (m n p )
12 8
18 12 30
 m n m n p
1218 812 30
m n p
30 20 30
m n p
( x  2 y)
6 4
2
8.
 ( x  2 y)
4  ( x  2 y)
( x  2 y)
6
 ( x  2 y)( x  2 y)
F O
I
L
 x  2 xy  2 xy  4y
2
2
 x  4 xy  4 y
2
2
6
5
ad
6  4 5 9
9. 4 9  a d  a 2 d 4
a d
2
a
 4
d