Multiplication and
Division of Exponents
Notes
Multiplication Rule
x x x
n
m
n m
In order to use this rule the base numbers
being multiplied must be the same.
Example:
3
x x
4
Written in multiplication form
x x x x x x x  x
Using Rule

x
3 4
x
7
7
Example 1
2 2
3
2*2*2
2
5
2*2*2*2*2
3 5
2
8

Example 2
x x x
354
3
5
x
x
12
4
Example 3
x x  y  y z
4
3
Remember


x
14
y
34
x 5 y 7 z
4
xx
z
1
Example 4
3
5
3
2
(2x y )(3xy )(x y)
Multiply coefficients and add exponents of like bases
6x

(312) (531)
y
6x y
6

9
Example 5
4x (2x y  5xy )
5
2
2
In order to simplify you must distribute. Since you are
multiplying when you distribute you must use the
multiplication rule for exponents

8x
52
y  20x
51 2
8x y  20x y
7
6
y
2

You Try
Simplify each expression
1.
y y
4
y
2
15 x 3
2
2.
3,
4.
5.
6.
6
(3x )(5x)
7
x
x x

5 6
3 
2
2 4
ab
(a b )(a b )
3
4
2

5
2
12y 3
(4 y )(3y)
4
6
(x y )(x y )


x9y8


Simplify (remember when
adding only add coefficients of
like terms)
1.
2.
3.
3x 3 (2x 2  5x  2)
8x 2  2x
2x(4x 1)

4 x(x  5)  (x 2  6x  8)


4.
6x 5 15 x 4  6x 3
4 x 5 (2x 3  6x) 12 x 6

4x 2  20x  x 2  6x  8
5x 2  26x  8
8x 8  24x 6 12x 6
8x 8  36x 6
Division Rule: If the bases are the
same subtract the exponents
m
x
mn
x
n
x
x 5 x x x x x
2


x
x3
x x x


5
OR
x
5 3
2

x

x
x3
Always do top
exponent minus
bottom exponent
Examples
1.
2.



3.
x2y6
xy 2
6a 7b 3
2ab 2


x5  x8
x3

x 21 y 62  xy 4
3a 71b 3 2  3a 6b
Divide coefficients,
subtract exponents of
the like bases.
x 5 8 x13
133
10
Use multiplication rule


x

x
3
3
x
x
on top, then use
division rule
Special Cases ( zero power):
any base raised to a power of
zero equals 1
x 1
0
Here is why, when the number in the numerator is the same as
the number in the denominator, the quotient is always 1.
2 4 2 * 2 * 2 * 2

1
4
2
2*2*2*2
So it makes
sense that
24
44
0

2

2
1
4
2
Examples
Simplify
3 4
1.
2.

ab
3
ab
a
10x 4 y 3
5x 4 y 2
2x 44 y 32  2x 0 y1  2*1y  2y
3 3 41
b
 a b  1b  b
0 3
3
3



3.
m10 n 5

10 5
m n

1010
m
n
5 5
 m n  1*1  1
0
0