why does a^0 = 1?

Question

I need specific stuff, not, ''Just becuase''. Like examples

Answer 1

WOW how you people stydy maths. You can write pages on it with knowing these 3 lines.

a^0 =1 proof

a^[0] = a^[+1 - 1] = a^[1] * a^[- 1]

as powers add up alzebraically.

= a^[+1] * {1/a^[+1] } because x^-1 = 1/x

= a^[+1]
---------------- = 1 (ONE)
a^[+1]

[Anything]^0 = 1 (any not=0) wife qualifies to be anything

Answer 2

Like many other people have mentioned, a^0 = 1 is partly just a definition, and partly from seeing what happens to a^n as n gets closer and closer to 0.

I just wanted to point out that 0^0 ist a bit of an exception. Obviously 0^1 and 0^2 and 0^3 are all equal to 0. 0^-1 is undefined. This is the same as asking what 1/0 is. So if you come from negative numbers, you would have to conclude that 0^0 is undefined. If you came from positive numbers, you would probably conclude that 0^0 equals 0.

But you could do this all differently too. You could watch what happens to a^0 as a gets closer to 0. Because a^0 = 1 for all values of a except 0 (which is what we are trying to find), we might want to conclude that 0^0 = 1.

So there you have it. 0^0 can equal 0, or 1, or is undefined; it all depends on what you are doing and what is convenient. This was not the main point of your question, but I thought it might round out an interesting special case.

Answer 3

becuase a^0 is just like
okay i will give you an example
let a=2
sub. a=2 into the equation
it will be like this
2^0=1
power 0 means the number of times u multiply the original number itself

just remember anything to the power of 0 is equal to 1
except for the number 0

Answer 4

I'll make a pattern before I explain.
10^3 = 1000
10^2 = 100
10^1 = 10
10^0 = 1
10^-1 = .1
10^-2 = .01
10^-3 = .001

When expressing a number exponentially, you're really expressing a factor of a number times a base number raised to some number.

Example:
8635 = 8.635 x 10^3
153 = 1.53 x 10^2
33 = 3.3^ 10^1
1 = 1 x 10^0

So, when you're expressing any number raised to the power of zero, all you're left with is the factor of that number. If you have no factor, you're answer is always 1. That even includes 0^0.

Answer 5

You know that a^m/a^n = a^m-n.

So, take a/a. It's the same as a^1/a^1
It equals a^1-1 = a^0. But obviously a/a=1.
So a^0 = 1.

I'll post if I can think of a more refined way.

Answer 6

Basically, it's by definition; but it's necessary to achieve consistency.

If you multiply x^2 and x^4, you end up with x^(2+4) = x^6.

If you multiply x^2 and x^0, you should end up with x^(2+0) = x^2. The only way this is consistent is if the zeroth power is defined as 1.

Answer 7

It's a mathematical rule. Anything raised to the 0 power will result in "1". Similiar to anything divided by itself will equal "1".

Answer 8

a^b/a^c = a^(b-c)

When b=c:

a^(b-c)=a^0

but then also a^b/a^c = 1

hence a^0 = 1 (except when a=0 is undefined)