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# Why is any number raised to the 0 power equal to 1?

## It is commonly taught that any number raised to the 0 power equal to 1 why is it so lets see, Any number raised to the 0 power equal to 1, can be explained with these methods,

 1) 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:Advertisement ```..., 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. 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^0 = x^(4-4) = --- = 1 x^4 ``` 3) Here’s another explanation from Aldo Daniel Completa, a contributor from Argentina:In the FAQ there is an explanation about ” n^0 (any number to zero power) .” I think another is this: Let’s begin with examples. ``` Ex.1: 5^3 / 5^2 = 125 / 25 = 5 5^3 / 5^2 = 5 ^(3-2) = 5^1 = 5 Ex.2: 5^3 / 5^3 = 125 / 125 = 1 5^3 / 5^3 = 5 ^(3-3) = 5^0 = ``` … and the result must be 1. So 5^0 =1.The rule is: x^b / x^c = x^(b-c). In order to generalize this rule for the case b=c, it must be defined that x^0 = 1 (x is any number different from 0). In mathematics, usefulness and consistency are very important. This convention allows us to extend definitions of power that would otherwise require treating 0 as a special case. This method can also explain the definition: x^(-b) = 1 / x^b. ``` Ex. 5^2 / 5^4 = 25 / 625 = 1 / 5^2 5^2 / 5^4 = 5 ^(2-4) = 5^(-2)```

Source: http://mathforum.org/

### Nitish

My Parents call me Nishu, and I am registered in my death certificate as. . . Oh sorry, in my birth certificate as Nitish Andola. Friends Often Also Call Me as Andy. I hate people who act over smart. I don’t take lot of time to adjust with anybody. It’s my strength rather. I love to live with my friends. I love all those who feel comfortable with me and who make me feel comfortable with them… I always think positive, which have never helped me to achieve anything, but still it has become a habit. I forgive people soon, as I feel this life is too short and we have no time to prove others wrong. I love innovative things. And friends sometime call me as HiFi. I love to be Honest, and one thing I have observed about myself is, I am not a Fan, I believe there is good and bad in everyone and we need to embrace the good. I enjoy the company of people who are FREE. I too love to live freely! I like to interact with nature and am very much interested in adventure sports and fascinated by Astronomy. I respect each and every person on this earth, sometimes even things and like to be a free spirit. Some don't like that, but that's the way I am. I never think badly for anyone...

### One Comment

1. Anonymous says:

Thank You…