Rational exp
onents
Summary of formulas involving general integer exponents
:
Before considering rational exponents it is appropriate to recall definitions and properties connected with
integer exponents.
The following formulas define integer powers of a nonzero real number
a
.
=
a
n
a . a . . . . a , n factors
=
a
(
)

n
1
a
n
=
a
0
1
where
n
is a positive integer.
The following formulas in which
m
and
n
are
any
integers follow from the definitions.
=
(
)
a b
n
a
n
b
n
=
a
b
n
a
n
b
n
=
a
m
a
n
a
(
)
+
m
n
=
a
m
a
n
a
(
)

m
n
=
(
)
a
m
n
a
(
)
m n
Rational number exponents
.
We try to deduce a sensible definition for
a
r
, where
a
is a nonnegative real number and
r
is a general rational number, that is,
=
r
p
q
,
where
p
and
q
are integers and
≠
q
0.
Our aim is to define
a
r
in such a way that it retains the same meaning as before in the case that
r
happens to be an integer, and also such the sum rule for exponents
=
a
r
. a
s
a
(
)
+
r
s
still applies when
r
and
s
are general rational numbers.
If the sum rule for exponents rule applies when
=
r
s
=
1
2
, then
=
a
1
2
. a
1
2
a
+
1
2
1
2
=
a
(
)
1
=
a
.
Thus, in order for the sum rule for rational exponents to apply here,
a
1
2
must be a real number with the property that, when it is
multiplied with itself, it gives the value
a
.
This means that we should identify
a
1
2
with the
square root
of
a
, that is, we should make the
definition
=
a
1
2
a
.
_____
In a similar way, using the sum rule,
=
a
1
3
. a
1
3
. a
1
3
a
+
+
1
3
1
3
1
3
=
a
(
)
1
=
a
.
Thus, in order for the sum rule for rational exponents to apply here,
a
1
3
must be a real number with the property that, when it is
multiplied with itself three times, it gives the value
a
. This means that we should identify
a
1
3
with the
cube root
of
a
, that is, we should
make the definition
=
a
1
3
3
a
.
_____
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In general, if
n
is a positive integer which is greater than or equal to 2, then, if the sum rule for rational exponents is to apply, we need to
have:
a
1
n
.
a
1
n
.
. . .
.
a
1
n
\___________________/
n
factors
=
a
+
+
+
1
n
1
n
. . .
1
n
, with
n
terms in the exponent.
=
a
(
)
1
=
a
.
Thus, in order for the sum rule for rational exponents to apply,
a
1
n
must be a real number with the property that, when it is multiplied
with itself
n
times, it gives the value
a
. This means that we should define
a
1
n
to be the
n
th
root of
a
, that is,
=
a
1
n
n
a
.
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 Spring '08
 Uri
 Exponents, Formulas, Negative and nonnegative numbers, 1

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