In Sample Set C, the exponents of the numerators were greater than the exponents of the denominators. Let’s study the case when the exponents are the same.

When the exponents are the same, say
n
n
, the subtraction
n−n
n−n
produces 0.

Thus, by the second rule of exponents,
x
n
x
n
=
x
n−n
=
x
0
x
n
x
n
=
x
n−n
=
x
0
.

But what real number, if any, does
x
0
x
0
represent? Let’s think for a moment about our experience with division in arithmetic. We know that any nonzero number divided by itself is one.

8
8
=1,
43
43
=1,
258
258
=1
8
8
=1,
43
43
=1,
258
258
=1

Since the letter
x
x
represents some nonzero real number, so does
x
n
x
n
. Thus,
x
n
x
n
x
n
x
n

represents some nonzero real number divided by itself. Then
x
n
x
n
=1
x
n
x
n
=1
.

But we have also established that if
x≠0,
x
n
x
n
=
x
0
x≠0,
x
n
x
n
=
x
0
. We now have that
x
n
x
n
=
x
0
x
n
x
n
=
x
0

and
x
n
x
n
=1
x
n
x
n
=1
. This implies that
x
0
=1,x≠0
x
0
=1,x≠0
.

Exponents can now be natural numbers and zero. We have enlarged our collection of numbers that can be used as exponents from the collection of natural numbers to the collection of whole numbers.

If
x≠0,
x
0
=1
x≠0,
x
0
=1

Any number, other than 0, raised to the power of 0, is 1.
0
0
0
0
has no meaning (it does not represent a number).

Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"