Simplify:
ⓐ
(
m
3
n
9
)
1
3
ⓑ
(
p
4
q
8
)
1
4 .
Solution
ⓐ
(
m
3
n
9
)
1
3
First we use the Product to a Power
Property.
(
m
3
)
1
3
(
n
9
)
1
3
To raise a power to a power, we multiply
the exponents.
m
n
3
ⓑ
(
p
4
q
8
)
1
4
First we use the Product to a Power
Property.
(
p
4
)
1
4
(
q
8
)
1
4
To raise a power to a power, we multiply
the exponents.
p
q
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We will use both the Product and Quotient Properties in the next example.
Simplify:
ⓐ
x
3
4
·
x
−
1
4
x
−
6
4
ⓑ
y
4
3
·
y
y
−
2
3 .
Solution
ⓐ
x
3
4
·
x
−
1
4
x
−
6
4
Use the Product Property in the numerator,
add the exponents.
x
2
4
x
−
6
4
Use the Quotient Property, subtract the
exponents.
x
8
4
Simplify.
x
2
ⓑ
y
4
3
·
y
y
−
2
3
Use the Product Property in the numerator,
add the exponents.
y
7
3
y
−
2
3
Use the Quotient Property, subtract the
exponents.
y
9
3
Simplify.
y
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Key concepts
Summary of Exponent Properties
If
a
,
b are real numbers and
m
,
n are rational numbers, then
Product Property
a
m
·
a
n
=
a
m
+
n
Power Property
(
a
m
)
n
=
a
m
·
n
Product to a Power
(
a
b
)
m
=
a
m
b
m
Quotient Property :
a
m
a
n
=
a
m
−
n
,
a
≠
0
,
m
>
n
a
m
a
n
=
1
a
n
−
m
,
a
≠
0
,
n
>
m
Zero Exponent Definition
a
0
=
1 ,
a
≠
0
Quotient to a Power Property
(
a
b
)
m
=
a
m
b
m
,
b
≠
0
Practice makes perfect
Simplify Expressions with
a
1
n
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
a
m
n
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.