|Why is any nonzero number raised to the power of
zero equal to one?|
|Question Date: 2015-09-10|
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
So a1 = a
a2 = a * a
or if N is negative that it is divided N times, so
a(-2) = (1/a) * (1/a) = 1 / (a2)
In this case, (a2 ) / (a2) =
(a * a) / (a * a) = 1
But when it comes to division, we also know that
aN / aM = a (N-M)
For example, a5 / a3 = ( a *
a * a * a * a ) / (a * a * a) = a * a =
a2 = a(5-3)
So, back to our previous example, a2 /
a2 = a(2-2) = a0
a2 / a2 =
(a * a )/(a * a) = 1
Thus, a0 = 1
So for example, when you go from 24 to
smaller powers, every time you divide by 2.
24 = 16
23 = Divide by 2 the 24 = 8
22 = Divide by 2 the 23 = 4
21 = Divide by 2 the 22 = 2
20 = ??? Divide by 2 the 21 = 1
2(-1) = Divide by 2 the 20 =
1/2 = 0.5
Hope these examples help.
A simple example to show this is that if you go
from 21 to 22 (2 to 4),
you multiply by 2.
The same is true if you go from 22
to 23 (4 to 8).
To go in the other direction, you would
divide by 2. So to go from 23 to
22, you would divide by 2. So if you
want to go to 21 to 20, 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.
This is a good question. If you take any number
and divide it by itself, the answer you get will
3/3 = 1 or 9/9 = 1
This applies to *all* numbers, even numbers raised
to a power. For example:
32/32 = 1
This makes sense because 32 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:
34/32 = 3(4-2) =
32 = 9
This makes sense because 34 is
really just 3*3*3*3, and 32 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:
34/32 = (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:
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
32/32 = 3(2-2) =
And we already know that the answer to
32/32 is 1, so we can
30 = 1
This is exactly what we set out to prove! That
30 = 32/32 = 1
Now you can try this same exercise with any
number at all and you will get the same answer.
It's because of the way that negative powers work.
Think about this:
21 = 2,
22 = 4,
2-1 = 1/2,
and 2-2 = 1/4.
20 is no multiplication or division
by 2 whatsoever. The only number that you could
multiply by 2n and actually get
2n is 1. Therefore, 20 = 1.
This same reasoning works for any other
non-zero number. Try it with 3 and see for
Any time we use powers, we can rewrite them as
division, like this: n(x-y) = nx/ny.
For example, 32 = 33-1
= 27/3 = 9.
If we apply this to the case of a zero power, we
30 = 31-1
= 31/31 = 1!
We can make this more general and say that
any number (n) with power of zero is equal to
n/n = 1.
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, 24 = 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, 23 x 21 =
2(3+1) = (2 x 2 x 2) x 2 =
24 = 16.
Now, we have what we need to explain why a base
(non-zero) to the zero-th power must be one, even
if imagining multiplying something zero is
confusing. Let’s go to our initial example:
24 can also be expressed as
24 x 20 because it is also
equal to 2(4+0) = (2 x 2 x 2 x 2) = 24.
The only number that can multiply a non-zero
number without changing its value is
Another way to think about it is to start with
20, which is equal to 2(4-4)
= 24 x 2-4. From before, we
know that 24 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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