Radical Expressions And Rational Exponents Worksheet Page 2
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1
2
3
4The properties of square roots in Lesson 1-3 also apply to nth roots.
Properties of n th Roots
For a > 0 and b > 0,
WORDS
NUMBERS
ALGEBRA
Product Property of Roots
The nth root of a product is equal
√ 16 =
√ 8 ·
√ 2 = 2
√ 2
√ ab =
n
√ b
n
3
3
3
3
a ·
√
n
to the product of the nth roots.
Quotient Property of Roots
The nth root of a quotient is equal
√ 25
√
n
√
√
a
25
= 5 _
a _
_
_
_
=
n
=
to the quotient of the nth roots.
√ 16
√ b
16
4
b
n
E X A M P L E
2
Simplifying Radical Expressions
Simplify each expression. Assume that all variables are positive.
√
_
3
√
3
x
3
A
6
B
27 x
7
√
3
3
x
√
3
_
· x
· x
3
3
3
Factor into
3
Quotient Property
√
3
7
perfect cubes.
When an expression
contains a radical
x _
√
√
√
3
3
3
·
·
3
3
3
Simplify the
3
x
x
Product
in the denominator,
√
3
7
numerator.
Property
you must rationalize
the denominator. To
_
_
√ 7
√ 7
3
3
3 · x · x
x _
Simplify.
·
·
Rationalize the
do so, rewrite the
√
3
√ 7
3
√ 7
3
7
expression so that
denominator.
the denominator
2
3x
√
3
2
contains no radicals.
x
7
_
Product Property
√
3
3
7
√ 49
3
x
_
Simplify.
7
Simplify each expression. Assume that all variables are positive.
√
8
x
√
4
√
√
_
4
3
3
·
4
7
2
2a.
16 x
2b.
2c.
x
x
3
__
m
A rational exponent is an exponent that can be expressed as
, where m
n
and n are integers and n ≠ 0. Radical expressions can be written by using
rational exponents.
Rational Exponents
For any natural number n and integer m,
WORDS
NUMBERS
ALGEBRA
__
1
1 _
1 _
The exponent
indicates the nth
√ 16 = 2
=
4
=
n
4
n
√
n
16
a
a
root.
__
m
2 _
(
)
m _
The exponent
indicates the nth
2
(
)
m
√ 8
√ a
3
n
=
= 2
= 4
=
=
n
3
2
n
√
n
m
8
a
a
root raised to the mth power.
8- 6 Radical Expressions and Radical Exponents
611
611
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