how anything power to zero is equal to one?

Answer 1

Observe this pattern:

3^2 = 9
3^1 = 3
3^0 = 1
3^(-1) = 1/3
3^(-2) = 1/9

Each time you divide by 3.

Answer 2

While the above argument might help convince your intuitive side that any number to the zero power is 1, the following argument is a little more rigorous.
This proof uses the laws of exponents. One of the laws of exponents is:

n^x
--- = n^(x-y)
n^y

for all n, x, and y. So for example,
3^4
--- = 3^(4-2) = 3^2
3^2


3^4
--- = 3^(4-3) = 3^1
3^3


Now suppose we have the fraction:
3^4
---
3^4

This fraction equals 1, because the numerator and the denominator are the same. If we apply the law of exponents, we get:
3^4
1 = --- = 3^(4-4) = 3^0
3^4

So 3^0 = 1.
We can plug in any in number in the place of three, and that number raised to the zero power will still be 1. In fact, the whole proof works if we just plug in x for 3:

x^4
---- = x^(4-4) = x^0 = 1
x^4

Answer 3

In general:
n^x = n^(x-y)*n^y, where n=!0 AND x=!0 (n and x are NOT both equal to zero).

In the particular case where x=y:
n^x = n^(x-x)*n^x => n^x = n^0*n^x
Divide by n^x gives 1 = n^0*1 => n^0 = 1

Example:
n=3, x=2, y=2 => 3^2 = [3^(2-2)]*3^1 => 3^2 = 3^0*3^2 => 3^0 = 1

If n=0 and x=0, then n^x = 0^0 which is indeterminate.

Answer 4

Let us have the example 1000

1000^1 = 1000

1000^(- 1) = 1 / 1000 = 0.001

0 lies between 1 & - 1.

1000 x 0.001 = ?

1000 x 0.001 = 1000^1 x 1000^(- 1)

1 = 1000^(1 - 1) Isn't it?

1 = 1000^0

Is that clear? You can apply this method to any natural number as base like 1000 and +ve and -ve forms of any number as two powers by placing the power 0 at an equal distance from either number e.g. n^m and n^(- m) where "n" is any natural number N = 1, 2, 3, 4, 5, 6, 7, 8, 9, ... and "m" may or may not vary from "n". All the best !

Answer 5

It is not "anything" to the power of zero that is equal to one, rather "finite" numbers, i.e. numbers that can be conceptualised. Thus 0^0 is indeterminate; so also is (inf)^0.
Imagine the finite number X raised to power a, b respectively
X^a/ X^b can also be written as X^(a-b), whilst X^a * X^b can also be written as X^(a+b).
For a = b, 1 = X^a/ X^b = X^(a - b) = X^0 QED

Answer 6

Hi
2 raised to power 2 means 2 is multiplied 2 times..i.e 2 * 2=4
2 raised to power 1 means 2 is multiplied 1 time..i.e 2 *[nothing]=2
so when 2 is raised to 0 means 2 is multiplied 0 times i.e it is not multiplied at all..and if u dont multiply any no. by any no. at all what does that mean??? it means 1.
Did u get that dear?

Answer 7

lets take an example x^n
we know anything divided by the same no. gives 1
so,x^n/x^n=1
according to the laws of exponents x^n/x^n=x^(n-n)=x^0=1
Hence anything raised to the power 0 is 1.

Answer 8

Its a rule of mathematics.Just the way 1 is neither a prime nor a composite number though it ought to be prime.But its a rule and you won't get the correct answer ever if you don't follow it.

Answer 9

Ha..Ha.....
It is impossible......
mathematics is a science......So obey the rules........