Simplifying
Exponents
1
Review Multiplication
Properties of Exponents
•  Product of Powers Property—To multiply powers
that have the same base, ADD the exponents.
•  Power of a Power Property—To find a power of a
power, multiply the exponents.
•  Power of a Product Property—To find a power of a
product, find the power of each factor and
multiply.
2
ANY NUMBER RAISED TO THE FIRST
POWER IS ITSELF.
a =a
1
FOR EXAMPLE:
3 =3
1
NOW YOU TRY:
528921 = 528921
1
3
Zero Exponents
•  Any number, besides zero, to the
zero power is 1.
•  Example:
•  Example:
0
a =1
0
4 =1
4
Negative Exponents
•  To make a negative
exponent a positive
exponent, write it as
its reciprocal.
•  In other words, when
faced with a negative
exponent—make it
happy by “flipping” it.
5
Negative Exponent
Examples
•  Example of Negative
Exponent in the
Numerator:
•  The negative exponent
is in the numerator—
to make it positive, I
“flipped” it to the
denominator.
1
x = 3
x
−3
6
Negative Exponents
Example
•  Negative Exponent in
the Denominator:
•  The negative exponent
is in the denominator,
so I “flipped” it to the
numerator to make
the exponent positive.
4
1
y
4
=
=
y
−4
y
1
7
Practice Making Negative
Exponents Positive
−3
1.  Try:
d
2.  Try:
1
−5
z
8
Answers to Negative
Exponents Practice
1.  Answer:
2.  Answer:
d
−3
1
= 3
d
5
1
z
5
=
=z
−5
z
1
9
Rewrite the Expression
with Positive Exponents
•  Example:
2x
−3
y
−2
•  Look at EACH factor and decide if the factor belongs in the
numerator or denominator.
•  All three factors are in the numerator. The 2 has a positive
exponent, so it remains in the numerator, the x has a
negative exponent, so we “flip” it to the denominator. The y
has a negative exponent, so we “flip” it to the denominator.
−3
2x y
−2
2
=
xy
10
Rewrite the Expression
with Positive Exponents
−
3
3
−
8
•  Example:
4 ab c
• 
All the factors are in the numerator.
Now look at each factor and decide if the
exponent is positive or negative. If the
exponent is negative, we will flip the
factor to make the exponent positive.
11
Rewriting the Expression
with Positive Exponents
•  Example:
−3
3 −8
4 ab c
•  The 4 has a negative exponent so to make the exponent positive—
flip it to the denominator.
•  The exponent of a is 1, and the exponent of b is 3—both positive
exponents, so they will remain in the numerator.
•  The exponent of c is negative so we will flip c from the numerator
to the denominator to make the exponent positive.
3
3
ab
ab
=
3 8
8
4c
64c
12
Practice Rewriting the
Expressions with Positive
Exponents:
1.  Try:
2.  Try:
−1
−2
−3
3 x y z
−2 3 −4
4a b c d
13
Answers
1.  Answer
2.  Answer
−1
3
4a
x
−2
−2
3
y
b c
−3
−4
z=
z
3x 2 y 3
3
4b d
d = 2 4
a c
14
Division Properties of
Exponents
•  Quotient of Powers Property
•  Power of a Quotient Property
15
Quotient of Powers
Property
•  To divide powers
that have the same
base, subtract the
exponents.
•  Example:
5
5−3
x
x
2
=
=
x
3
x
1
16
Practice Quotient of
Powers Property
1.  Try:
9
a
3
a
3
2.  Try:
y
4
y
17
Answers
1.  Answer:
9
9 −3
a
a
=
3
a
1
= a6
3
2.  Answer:
y
1
1
= 4 −3 =
4
y
y
y
18
Power of a Quotient
Property
•  To find a power of a
quotient, find the
power of the
numerator and the
power of the
denominator and
divide.
•  Example:
⎛a⎞
⎜
⎟
⎝b ⎠
3
3
a
=
3
b
19
Simplifying Expressions
3 4
•  Simplify
⎛ 2m n
⎜⎜
⎝ 3mn
⎞
⎟⎟
⎠
3
20
Simplifying Expressions
•  First use the Power of a Quotient Property
along with the Power of a Power Property
3
⎛ 2m n
⎜⎜
⎝ 3mn
4
3
3
3⋅3
4⋅3
3
9
12
⎞
2 m n
2 mn
⎟⎟ = 3 3 3 = 3 3 3
3 mn
3 mn
⎠
21
Simplify Expressions
•  Now use the
Quotient of Power
Property
3
9 12
9−3 12−3
2 mn
8m n
=
3 3 3
3mn
27
6
8m n
=
27
9
22