
Why is any nonzero number raised to the power of
zero equal to one?   Answer 1:
Let's first agree on some conventions. For
example, that some number a raised to the
power of N, where N is an integer (such as
3, 105, 23) meansa * a * a * a multiplied N
times.
So a^{1} = a
and
a^{2} = a * a
or if N is negative that it is divided N times, so
a^{(2}) = (1/a) * (1/a) = 1 / (a^{2})
In this case, (a^{2} ) / (a^{2}) =
(a * a) / (a * a) = 1
But when it comes to division, we also know that
a^{N} / a^{M} = a^{ (NM)}
For example, a^{5} / a^{3} = ( a *
a * a * a * a ) / (a * a * a) = a * a =
a^{2} = a^{(53)}
So, back to our previous example, a^{2} /
a^{2} = a^{(22)} = a^{0}
but also,
a^{2} / a^{2} =
(a * a )/(a * a) = 1
Thus, a^{0} = 1
So for example, when you go from 2^{4} to
smaller powers, every time you divide by 2.
2^{4} = 16
2^{3} = Divide by 2 the 2^{4} = 8
2^{2} = Divide by 2 the 2^{3} = 4
2^{1} = Divide by 2 the 2^{2} = 2
2^{0} = ??? Divide by 2 the 2^{1} = 1
2^{(1)} = Divide by 2 the 2^{0} =
1/2 = 0.5
Hope these examples help.
  Answer 2:
A simple example to show this is that if you go
from 2^{1} to 2^{2} (2 to 4),
you multiply by 2.
The same is true if you go from 2^{2}
to 2^{3} (4 to 8).
To go in the other direction, you would
divide by 2. So to go from 2^{3} to
2^{2}, you would divide by 2. So if you
want to go to 2^{1} to 2^{0}, you
would also divide by 2. If you divide 2 by 2,
you get 1. The same is true of any number,
to get to the zero power, you divide the
number by itself which gives you 1.
  Answer 3:
This is a good question. If you take any number
and divide it by itself, the answer you get will
be 1.
For example:
3/3 = 1 or 9/9 = 1
This applies to *all* numbers, even numbers raised
to a power. For example:
3^{2}/3^{2} = 1
This makes sense because 3^{2} is just
3*3, which of course is just 9.
Now, one of the rules of powers says that as
long as the base number is the same, all you have
to do to divide two numbers raised to a power is
to just subtract the top power from the bottom
power,. like this:
3^{4}/3^{2} = 3^{(42)} =
3^{2} = 9
This makes sense because 3^{4} is
really just 3*3*3*3, and 3^{2} is just
3*3, so the question is really just asking:
(3*3*3*3)/(3*3) = ?
And we know from arithmetic rules that we can
cancel same numbers on the top and bottom of the
division and get:
3^{4}/3^{2} = (3*3*3*3)/(3*3) =
3*3 = 9
But notice that this rule only works when the
base number (3 in this case) is the same for both
the top and bottom.
What happens when the power on both the top
and bottom number is the same? For example:
3^{2}/3^{2}
We know from earlier that the answer will be
1, which is a lot like the answer you get when
you raise any number to the power of zero,
isn’t it? Let’s apply the rule of powers
from before:
3^{2}/3^{2} = 3^{(22)} =
3^{0}
And we already know that the answer to
3^{2}/3^{2} is 1, so we can
see that
3^{0} = 1
This is exactly what we set out to prove! That
is because
3^{0} = 3^{2}/3^{2} = 1
Now you can try this same exercise with any
number at all and you will get the same answer.
  Answer 4:
It's because of the way that negative powers work.
Think about this:
2^{1} = 2,
2^{2} = 4,
2^{1} = 1/2,
and 2^{2} = 1/4.
2^{0} is no multiplication or division
by 2 whatsoever. The only number that you could
multiply by 2^{n} and actually get
2^{n} is 1. Therefore, 2^{0} = 1.
This same reasoning works for any other
nonzero number. Try it with 3 and see for
yourself!
  Answer 5:
Any time we use powers, we can rewrite them as
division, like this: n(xy) = nx/ny.
For example, 3^{2} = 3^{31}
= 3^{3}/3^{1}
= 27/3 = 9.
If we apply this to the case of a zero power, we
can rewrite:
3^{0} = 3^{11}
= 3^{1}/3^{1} = 1!
We can make this more general and say that
any number (n) with power of zero is equal to
n/n = 1.
  Answer 6:
To answer this, we need to know some of the
properties that powers have. First, a
raised to the b power is equal to a
multiplied by itself b times. For a = 2 and
b = 4, 2^{4} = 2 x 2 x 2 x 2 = 16. This is
straightforward for a positive b. What
about a negative b?
A negative power is the reciprocal (or fraction
of one) of the positive value. For example, if
b = 4 in the example above: 2^{4}
= 1/(2 x 2 x 2 x 2) = 1/16
Second, we need to know that powers can be
added together if the base (a) is the same.
As examples, 2^{3} x 2^{1} =
2^{(3+1)} = (2 x 2 x 2) x 2 =
2^{4} = 16.
Now, we have what we need to explain why a base
(nonzero) to the zeroth power must be one, even
if imagining multiplying something zero is
confusing. Let’s go to our initial example:
2^{4} can also be expressed as
2^{4} x 2^{0} because it is also
equal to 2^{(4+0)} = (2 x 2 x 2 x 2) = 24.
The only number that can multiply a nonzero
number without changing its value is
one.
Another way to think about it is to start with
2^{0}, which is equal to 2^{(44)}
= 2^{4} x 2^{4}. From before, we
know that 2^{4} x 2^{4} = 16 x
(1/16) = 1. This is true for any nonzero number.
Things are tricky when you try to think about
zero to the zeroth power – it is considered
“undefined”, not a number at all.
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