anything to the power of zero equals to one?

Question

Can anyone technically explain why anything to the power of zero equals to one?...'techinically' or 'practically'. Thanks

Answer 1

Look at this list...
2^2 = 4
2^1 = 2
2^0 = x
2^-1 = 1/2
2^-2 = 1/4

1 is the geometric mean between 1/2 and 2.
1 is the geometric mean between 1/4 and 4.

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

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:


1 = (3^4)/(3^4)= 3^(4-4) = 3^0


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

Answer 4

it means that whatever x is, x^0=1. This makes sense because when you increase the exponent by one you are multiplying by x, and when you decrease the exponent by one you are dividing by x. So since x^1=x, x^0= x^1/x or x/x =1
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Look at this list...
2^2 = 4
2^1 = 2
2^0 = x
2^-1 = 1/2
2^-2 = 1/4

1 is the geometric mean between 1/2 and 2.
1 is the geometric mean between 1/4 and 4.

Answer 5

One reason is because of the following rule in exponents:
(a^b)/(a^c) = a^(b-c)

So if b and c are the same, we have a^b / a^b on the left side (which is just 1), and a^0 on the right side. So this works for all values of a or b. (Except for a=0, because 0^0 is what's called indetermined. Basically, you can't take 0 to the 0 power, just as you can't divide by 0.)

This site (which is where another user here got his answer without giving credit) also explains things in more detail:
http://mathforum.org/dr.math/faq/faq.number.to.0power.html

Answer 6

Anything, but 0, to the power of 0 is 1 because anything divided by itself, except for 0, is 1.

For example:
10^3=10*10*10=1000
10^2=10*10=100
10^1=10=10
10^0=10/10=1
10^-1=10/(10*10)=1/10

Alternative explanation: By the definition of exponents when you find something to the power of zero you divide it by itself.

Answer 7

The "power" of a variable is really a representative of how many times you multiply that variable. For example, x^2 = x(x). We know that x^1 = x because we multiply x by itself only once. In your question x^0 = 1 we multiply x by itself no times, thus getting a result of one.

This same rule applies to negative powers, except that rather than multiplying, we do the inverse operation: division. So x^-2 = 1/x(x) = 1/x^2.

Also, it is important to note that "anything to the zero power is one" only applies to the variable you are considering. For example, 2x^0 = 2(1) = 2.

Answer 8

Anything other than 0. 0^0 is undefined.

2^3=2*2*2*1
2^2=2*2*1
2^1=2*1
2^0=1

Answer 9

it means that whatever x is, x^0=1. This makes sense because when you increase the exponent by one you are multiplying by x, and when you decrease the exponent by one you are dividing by x. So since x^1=x, x^0= x^1/x or x/x =1

Answer 10

ecept zero itself. X^^1=X X^^0 = X^^1/X = X/X = 1