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1
Lesson 36
Section 7.2
What is a value for
2
1
9 ?
Consider This.
9
9
9
9
9
1
2
1
2
1
2
1
2
1
=
=
=
×
+
using the product rule of exponents.
Now, Think!
What other number times itself equals 9?
9
3
3
=
×
Since both products equal 9, we can conclude that
3
9
2
1
=
.
Conclusion:
A rational exponent is the same as a root!
Definition of a rational (fractional) exponent:
n
n
a
a
=
1
Evaluate or write an equivalent root for each.
Simplify, if possible.
1.
=
2
1
16
2.
=
3
1
x
3.
=

5
1
)
32
(
4.
=
4
1
)
4
(
c
Write an equivalent expression using exponential notation.
5.
=
4
5
6.
=
3
2
ab
7.
=
n
Expanded Definition of a rational (fractional) exponent:
( )
m
n
n
m
n
m
a
a
a
or
=
Proof:
( )
( )
1
1
m
m
m
n
n
n
m
m
m
n
n
n
a
a
a
a
a
a
=
=
=
=
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Notice:
The exponent
m
can be inside the radical (in the radicand) or outside of the
radical.
The order does not matter.
You can raise to the
m
power first and then take the
n
th root.
Or you can take the
n
th root and then raise to the
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