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In xn,
x is the base,
and n is the exponent
(or power)
We defined positive integer
powers by
xn
= x ·
x ·
x ·
. . . · x (n factors of x)
The above definition can
be extended by requiring other
powers (i.e. other than positive
integers) to behave like the
positive integer powers. For
example, we know that
xn
xm = xn
+ m
for positive integer powers,
because we can write out the
multiplication.
Example:
x2
x5
= (x · x)(x · x · x · x · x) = x · x · x · x · x · x · x = x7
We now require that this
rule hold even if n
and m are not positive
integers, although this means
that we can no longer write
out the multiplication (How
do you multiply something
by itself a negative number
of times? Or a fractional
number of times?).
We can find several new properties
of exponents by similarly
considering the rule for dividing
powers:
(We will assume without always
mentioning it that x
¹
0). This rule is quite reasonable
when m and n
are positive integers and
m > n. For
example:
where indeed 5 – 2 = 3.
However, in other cases it
leads to situation where we
have to define new properties
for exponents. First, suppose
that m < n.
We can simplify it by canceling
like factors as before:
But following our rule would
give
In order for these two results
to be consistent, it must
be true that
or, in general,
·
Notice that
a minus sign in the exponent
does not make the result negative—instead,
it makes it the reciprocal
of the result with the positive
exponent.
Now suppose that n
= m. The fraction becomes
 ,
which is obviously equal
to 1. But our rule gives
Again, in order to remain
consistent we have to say
that these two results are
equal, and so we define
x0
= 1
for all values of x
(except x = 0, because
00 is undefined)
The following properties
hold for all real numbers
x, y, n,
and m, with these exceptions:
1.
00 is undefined
2.
Dividing by zero is
undefined
3.
Raising negative numbers
to fractional powers can be
undefined
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x1
= x |
(xn)m
= xnm |
|
x0
= 1 |
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xn
xm
= xn +
m |
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