Can somebody prove for me why any number to power zero equals 1?

Answer 1

Use the laws of exponents.

One law of exponents is
(n^x) / (n^y) = n^(x-y)

So consider
n^0
This is the same as
n^(a-a), for any a

Which can be written, by using the law of exponents as
(n^a) / (n^a) for any a, as long as n is not 0.

And we know that anything divided by itself is always 1.

Answer 2

Anything to the power zero is just "1". There are various explanations. Tracing through a progression using the number 3 as an example:
3^1 = 3

3^5 = 3^6 ÷ 3^1 = 243
this could be rewritten 3^(6-1) or 3^5 and so on
3^4 = 3^5 ÷ 3 = 81
3^3 = 3^4 ÷ 3 = 27
3^2 = 3^3 ÷ 3 = 9
3^1 = 3^2 ÷ 3 = 3

Then logically
3^0 = 3^1 ÷ 3 = 3 ÷ 3 = 1.

Using x as a representative for any number
x^0 = x^1 ÷ x = x/x = 1

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A negative-exponents explanation might be as follows:
x^0 = x^(n – n) = x^n × x^(–n) = x^n ÷ x^n = 1
...since anything divided by itself is just "1".

Answer 3

You must be knowing that any number divided my itself is 1. Say 2/2 = 1 then .. 3/3 = 1 and so on .. Now according to the rules of the power, when u want to divide the same number with different power, u substract the power of the denominator frm the power of numerator - To illustrate this : (8^5)/(8^3) = 8^(5-3) which makes it 8^2 = 64 Now consider a sum like 8/8 - you know the answer's 1 but it can also be expressed as 8^1/8^1 (because any number with power 1 is the number itself !) So according to the rules - 8^1/8^1 = 8^(1-1) = 8^0 = 1 (because 8/8 = 1) In this way it applies to any number ! Moreover, any number to the zero power means the number is being divided by itself.

Answer 4

We know that x^m/x^n = x^(m-n)
Thus x^m/ x^m = x^0 = 1 because anything divided by itself is 1.
Note: 0/0 is indeterminate and could be an exception.

Answer 5

Another way is to look at the limit of n^x as x approaches 0. Play with a calculator - try 3^0.1, 3^0.001; 3^-0.01, 3^-0.0001 ... Have fun.

Answer 6

it is a definition and has no explanation