Math question from a curious student!?

Question

why is it that any number to the zeroth power is always one?

Answer 1

Let x be any number.

We want to find out what is x^0.

Notice that:

x^0 = x ^ (y - y)

since y - y = 0, where y is any number.

Then

x^(y -y) = (x^y) * (x^-y) = (x^y) * 1/(x^y)

= (x^y)/(x^y)
=1

Hence, x^0 = 1

That's why any number to the zeroth powre is always one

Answer 2

Very simply the exponent tells how many of the number to multiply togeter. => 1*1*1 = 1^3 = 1
If the exponent is 0 then you are multiplying none of the numbers together.
In practice, you are multiplying the exponential to another number, for example:
4 * 5^0
If you read the sentence, it says what do I get if I have 4 and don't multipy it by any 5's. Well if you don't multiply by anything you still have 4. Therefore in theory 5^0 = 1. That is 4 * 1 = 4.

Zero on the other hand is a very powerful number in mathematics. Since 0 to any non-zero number is alway zero and any non-zero number to the power zero is 1, is 0^0 = 1 or 0.
This is a big problem for us, so we just say that it is undefined or simply it is allowed to be used in an equation.

Answer 3

I figured this out once, hang on...because when you put a number to the power of nothing, then you have to come out with a number which, no matter what expoenent you "put it" to does not change, that leaves you with 1 and 0, but anything plus zero does not change the first number, so they had to say 1.

WTF am I saying? I'm full of it on this one I can't remember the actual answer.

Answer 4

think of it this way

5^3=125
5^2=25 125/5=25
5^1=5 25/5=5
5^0=1 5/5=1
5^-1=1/5 1/5=1/5

in expanded form
5*5*5=125
5*5=25 125/5= 25
5*1=5 (5^1) 25/5= 5
its just dividing by 5
5^0=1

another example

4^3=64
4^2=16
4^1=4
4^0=1
dividing by 4
this is a late answer.. hope i helped

Answer 5

What the people have said so far is almost completely correct, but there is one error.

0^0≠1

The proof involves x^0=x^(a-a)=x^a/(x^a)=1. Well, if x=0 then we can't do the 0^a/(0^a) step because division by zero is always undefined.

Kudos to the previous answers, I just wanted to clarify things.

Answer 6

Eulercrosser is correct. Another way of realising this is using limits. Say, a not zero, as x -> 0, a^x -> 1.

Answer 7

lets use 3 as an example

(3^2) / (3^2) is 3^(2-2) using index laws
3^(2-2) is 3^0
when you divide something by itself (in this case it's 3^2) you always get one.

soooooo 3^0 = 1

Answer 8

if we suppose a nuber A
and we put it to power of n
so A ^ n = A*A*A*...... for n times
so when
A^0 = A ^ (n-n)
A (n-n) = A^n / A^n = 1
so any number to the power zero must = 1
Note : 0^0 is undefined value.

Answer 9

Because one is the first number.

Answer 10

The following link gives a pretty good answer to your question. Hope it helps.
http://mathforum.org/library/drmath/view/62701.html