
Why any number to the zero power always gives a one?

Question Date: 20110901   Answer 1:
This is an excellent question! There are lots
of different ways to think about it, but here's
one: let's go back and think about what a power
means. When we raise a number to the nth power,
that really means that we multiply that number by
itself n times, so for example, 2^{2} =
2*2 = 4, 2^{3} = 2*2*2 = 8, 3^{4}
= 3*3*3*3 = 81, and so on. So when we raise a
number to the zeroth power, that means we multiply
the number by itself zero times  but that means
we're not multiplying anything at all! What does
that mean? Well, let's go even farther back to
the simplest case: addition. What happens when we
add no numbers at all? Well, we'd expect to get
zero, because we're not adding anything at all.
But zero is a very special number in addition:
it's called the additive identity, because it's
the only number which you can add to any other
number and leave the other number the same. In
short, 0 is the only number such that for any
number x, x + 0 = x. So, by this reasoning, it
makes sense that if adding no numbers at all gives
back the additive identity, multiplying no numbers
at all should give the multiplicative identity.
Now, what's the multiplicative identity? Well,
it's the only number which can be multiplied by
any other number without changing that other
number. In short, the multiplicative identity is
the number 1, because for any other number x, 1*x
= x. So, the reason that any number to the zero
power is one is because any number to the zero
power is just the product of no numbers at all,
which is the multiplicative identity, 1.
  Answer 2:
It's exciting to me that you asked this
question. The fact is these rules are presented as
somewhat arbitrary, but there is always, always
(well almost always) good reason for them. Keep it
up! If it ever sounds arbitrary then hound your
teacher. If your teacher can't give you compelling
reasons why something is true, hound us or hound
Google. Okay, enough, onto your question:
Mathematics was initially developed to describe
relationships between everyday quantities
(generally whole numbers) so the best way to think
about powers like a^{b} ('a' raised to the
'b' power) is that the answer represents the
number of ways you can arrange sets of 'b' numbers
from 1 to 'a'. For example, 2^{3} is 8.
Why? There are 8 ways to write sets of 3 numbers
where each number can be either 1 or 2:(1,1,1)
(1,1,2) (1,2,1) (2,1,1) (2,1,2) (2,2,2) (1,2,2)
(2,2,1). So what does 3^{0}
represent? It is the number of ways you can
arrange the numbers 1,2, and 3 into lists
containing none of them! How many ways are there
to place a penny, a nickel, and a quarter on the
table such that no coins are on the table? Just
one... don't put anything on the table and that's
your only option. Therefore it's consistent to say
3^{0} = 1. There are other reasons
why a^{0} has to be 1  for example, you
may have heard the power rule: a^{(b+c)} =
a^{b} * a^{c}. What happens if b =
0? Well I know that a^{(0+c)} is the same
as a^{c} and by the first formula this is
also a^{0} * a^{c}. What choice
for a^{0} make sense to satisfy
a^{c} = a^{0} * a^{c}?
Clearly a^{0} = 1. I know this sounds a
little fishy since we started with a rule I could
have just made up (which is why I gave the other
reason first), but these formulas are all
consistent and there is never any magic step, I
promise! You can always work backwards to prove
a^{(b+c)} = a^{b} * a^{c}
etc etc. Hope this helps!
  Answer 3:
Any number to the zero power always gives
one. One rule for exponents is that
exponents add when you have the same base. So if
you have a number, x, and exponents, a and b,
then: x^{a} * x^{b} =
x^{(a+b)} So then if we make one of the
exponents negative:x^{a} * x^{b}
= x^{(ab)} And if the exponents are
the same magnitude (a = b)x^{a} *
x^{b} = x^{a} * x^{a} =
x^{(aa)} = x^{0} Now,
remember that if you have a negative exponent, it
means you have one divided by the number to the
exponent: x^{a} =
1/x^{a} So, we can also write
x^{a} * x^{a} in a different
way: x^{a} * x^{a} =
x^{a} * 1/x^{a} =
x^{a}/x^{a} And a number
divided by itself is always 1
so: x^{a} * x^{a} =
x^{a}* 1/x^{a} =
x^{a}/x^{a} = 1: So now we've
shown that: x^{a} * x^{a} =
x^{(aa)} =
x^{0} and x^{a} *
x^{a} = x^{a} *
1/x^{a}: This means that any number
x^{0} = 1.If you had trouble
understanding it all with variables, let's look at
it again,but this time as an example with
numbers: If we plug in numbers, (for example
let x = 5, a = 2, and b = 4) then: One rule
for exponents is that exponents add when you have
the same base. 5^{2} *
5^{4} = 5^{(2+4)} = 5^{6}
= 15625 So then, if we make one of the
exponents negative: 5^{2} *
5^{4} = 5^{(24)} =
5^{2} = 0:04 And if the exponents are
the same magnitude: 5^{2} *
5^{2} = 5^{(22)} =
5^{0} Now, remember that if you have
a negative exponent, it means you have one divided
by the number to the exponent: 5^{2} =
1/5^{2} = 0:04 So we can also write
5^{2} * 5^{2} in a different
way: 5^{2} * 5^{2} =
5^{2} * 1/5^{2} =
5^{2}/5^{2} = 25/25 And a
number divided by itself is always 1
so: 5^{2} * 5^{2} =
5^{2} * 1/5^{2} =
5^{2}/5^{2} = 25/25 = 1 So
now we've shown
that: 5^{2}*5^{2} =
5^{(22)} =
5^{0} and 5^{2} *
5^{2} = 5^{2}/5^{2} =
1 This means that 5^{0} = 1. This
works for any number x that you want to plug in
except for x = 0,because 0/0 is indeterminate (it
is like dividing zero by zero).
  Answer 4:
Let's look at what it means to raise a number
to a certain power: it means to multiply that
number by itself a certain number of times. Three
to the second power is three multiplied by itself
2 times, or 3*3=9. Let's look at a few
examples: 3^{5} = 3*3*3*3*3 =
243 3^{4} = 3*3*3*3 =
81 3^{3} = 3*3*3 = 27 3^{2}
= 3*3 = 9 3^{1} = 3 = 3 But how
do you go from 3^{1} to 3^{0}? If
you look at the pattern, you can see that each
time we reduce the power by 1 we divide the value
by 3. Using this pattern we can not only find the
value of 3^{0}, we can find the value of 3
raised to a negative power! Here are some
examples: 3^{0} = 3/3 =
1 3^{1} = 1/3 = 0.3333... (this
decimal repeats forever) 3^{(2)} =
1/3/3 = 0.1111... 3^{(3)} = 1/3/3/3 =
0.037037... No matter what number we use
when it is raised to the zero power it will always
be 1. Suppose instead of 3 we used some
number N, where N could even be a decimal.
N^{1}=N, and to reduce the power by 1 we
divide by N, soN^{0}=N^{1}/N =
N^{2}/2N =1. Notice that
3^{(1)} is the same as 1/(3^{1}),
3^{(2)} is the same as
1/(3^{(2)}),and so on. This gives us a
useful property of exponents, namely that
a^{(b)} is the same as 1/(a^{b}).
  Answer 5:
Heres a quick demonstration of why any number
(except zero) raised to the zero power must equal
1. As an example we will let that any number be
the number 3. Note that: 3^{1} =
3 = 3 3^{2} = 3*3 = 9 3^{3}
= 3*3*3 = 27 3^{4} = 3*3*3*3 =
81 And so on Youll notice that
3^{3}=(3^{4})/3,
3^{2}=(3^{3})/3,
3^{1}=(3^{2})/3 In other words,
3^{(n1)}=(3^{n})/3 So
3^{0}=(3^{1})/3=3/3=1 This
same reasoning will work for any number (not just
3), except the number 0. It wont work for 0
because you cant divide by 0. Lets call any number
x: x^{(n1)}=x^{n}/x So
x^{0} = x^{(11)} =
x^{1}/x = x/x = 1
  Answer 6:
of the answer is that this is how we've
defined powers to be.
Raising something to
a power greater than zero means multiplying it by
itself a number of times equal to the power. So,
for instance,
2^{1} = 2
2^{2} = 2 x 2 = 4 2^{3} = 2 x
2 x 2 = 8 and so on.
Now, you can
multiply anything by 1 and it will still be the
same thing, and likewise you can divide anything
by 1 and it will still be the same. Therefore:
2^{1} = 2 x 1 = 2
2^{2} = 2 x 2 x 1 = 4
2^{3} = 2 x 2 x 2 x 1 = 8
You see I've just multiplied everything
by 1.
Now, also note that if you raise
something to a negative power, then you take the
reciprocal of that something:
2^{1} = 1/2
2^{2} = 1 /(2x2) = 1/4
2^{3} = 1/8
And so
on. Again, we can multiply by everything by 1:
2^{1} = 1 x 1/2
Now, what happens
when the power is zero?
Well, you're not
multiplying by anything, except the 1 you started
with. You're not dividing by anything, except the
1 you started with. So, what you're left over with
is 1.
Now, here is the slightly more
mathematically sophisticated version: when you
raise something to a power, what you do is take 1
and multiply it by the base of the power a number
of times equal to the power. So, by definition,
raising something to the power of zero means you
start with 1, and then don't multiply it by
anything. So, naturally, 1 is what you're left
over with.
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