Why X power to 0 equal to 1 ?

Question

What's the physical meaning of :
  0
X  = 1

Answer 1

In multiplication, 1 is the identity -- that's like a default number. This is because you can ALWAYS say that 1 is a factor of ANY number. Exponentiation (power to) is a kind of multiplication, a short hand way of writing to multiply a number by itself a certain number of times. Since you are multiplying the number (x) by itself zero times, the only factor will be the 1. I hope this helps?

Answer 2

Throughout history, mathematicians have had to come up with new concepts when old ones proved insufficient. Negative numbers, for instance, were invented as a way to solve the problem, "How can I take 5 away from 3? The answer can't be zero, because then 3-5=0 and 3-6=0... we can't take two different numbers away from 3 and get the same result for each!"

The original notion of an exponent (or "power") was that it was a count of factors. x^2 meant two factors of x being multiplied together. But once the properties of exponents were worked out, there was a side effect. One of those properties was that, when dividing two exponential terms that had the same base, the result involved subtracting the powers. (For instance, x^5/x^2 = x^3.) And that led to the question: what about x^5/x^5? That should equal x^0. And how do you multiply no factors of something together?

The answer was to remember that anything divided by itself gives 1 as a quotient. x^5/x^5 = 1 because anything divided by itself is 1. Therefore, x^0 is *defined* to be equal to 1 -- because, in order for math to be consistent, there's really no choice.

Hope that helps!

Answer 3

Let's first look at an example. Let's look at the list of numbers

3^1, 3^2, 3^3, 3^4, ....
Finding the actual values, we get 3, 9, 27, 81, ....


So what is the pattern in the bottom sequence? Well, every time you move to the right in the list you multiply by 3, and every time you move to the left in the list you divide by 3. So we could take the bottom sequence and keep going to the left and dividing by 3, and we'd have the sequence that looks like this:

..., 3^-3, 3^-2, 3^-1, 3^0, 3^1, 3^2, 3^3, 3^4, ....

..., 1/27, 1/9, 1/3, 1, 3, 9, 27, 81, ....


So now we know what all the powers of 3 are! Actually, we just did the integer powers of 3. But that's probably enough for now.

There's a more detailed proof at the following link:

Answer 4

It is the pattern.

3^4 = 81 = 3 x 3 x 3 x 3
3^3 = 27 = 3 x 3 x 3
3^2 = 9 = 3 x 3
3^1 = 3 = 1 x 3
3^0 = 1

If x^0 = 0, you can't multiply it with anything. It would always be 0.
That is why it is equal to 1.

Answer 5

For a "physical meaning," try this:

Pick any nonzero integer.
Write its prime factorization.

For the sake of argument, I'll pick: 98784 = 2^5 * 3^2 * 7^3.

Let x = any number. Ask yourself, how many powers of x are in the factorization of 98784?

If x = 2, the number of powers is 5. 2^5 divides 98784.
If x = 3, the number is 2. 3^2 divides 98784.
If x = 4, the number is 2. 4^2 divides 98784.
If x = 5, the number of powers is 0. There are no powers of 5 in 98784.

x^0 power represents the identity, when x is not equal to zero.

Answer 6

First, remember that this is not true when x = 0.

Second, the easiest explanation is to apply the division rule. Remember that x^m/x^n = x^(m-n). That is, when I have a value raised to an exponent in the numerator and the same value raised to an exponent in the denominator, the result is achieved by subtracting exponents.

This works because of how things will "cancel" out. For example,
x6/x3 = (xxxxxx)/(xxx) = xxx = x^(6-3) = x^3. Notice that I had six x's in the top (x^6) and three in the bottom (x^3). Three of the top ones cancelled with three of the bottom ones, leaving three in the top.

This division rule will always work.

So what if I have x^3/x^3? Obviously the three x's in the top cancel with the three in the bottom, leaving me with 1 as a result. But the division rule also tells me that x^3/x^3 = x^(3-3) = x^0.

Therefore, x^0 = 1 because of the division rule of exponents.

Answer 7

Using laws of exponents

(x^n)/(x^n) = x^(n - n) = x^0

as you can see, you have the same value in the numerator and denominator, as as you learned, any number divided by itself will get you 1.

for ex:

x = 3
n = 2

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

therefore

3^0 = 1

Answer 8

x^0 = x^(n-n) = x^n/x^n = 1

Answer 9

If you multiply x^n by x, you obtain x^(n+1). So, the product of x^0 and x is x [= x^1]. If x is nonzero, x^0 must therefore be equal to 1

Answer 10

if x = 2
x^1 = x = 2
x^1 * x^1 = x^(1+1) = x^2 = 2*2 = 4
x^2 / x^1 = x^(2 -1) = x^1 = 4/2 = 2
x^1 / x^1 = x^(1-1) = x^0 = 2/2 = 1

this is true to any given value