Why does any thing to the power of zero equal one?

Question

I asked my math teacher and she wouldnt tell me i tried to go to a math dot com but i didnt understand im so confused!

Answer 1

0 to the 0 power is not one it's indeterminate, everything else is
If you're familiar with how exponents work you should know that:

x^n/x^m = x^(n-m) (ex: x^5/x^2 = x^(5-2) = x^3)

now suppose we take
x^3/x^3 = 1 =
x^3/x^3 = x^(3-3) = x^0

it works for any x^n/x^n = x^(n-n) = x^0 = 1

based on this, it makes sense that 0^0 wouldn't be 1 because there would be 0 in the denominator, which is not allowed

Answer 2

I believe it's a convention among mathematicians. However, it's a convention that makes sense, and here's why:
Start with the very defintion of a power:
x^n (ie x to the power n) means x*x*....x
That is x^n is x multiplied by itself n times.
Now if m and n are integers, then:
(x^n)*(x^m) = (x*...*x)*(x*...*x)= x*......*x = x^(n+m)

Now let n =1 and m =0
Then (x^n)*(x^m) = (x^1)*(x^0) = x^(1+0) = x^1 = x
This shows that for any nonzero number x, x*(x^0) = x
So x^0 must be 1 because 1 is the only number with the property x*y = x
That is, 1 is the only number that when multiplied by any nonzero number gives that nonzero number back.

Answer 3

For starters, not anything to the power of zero des equal one.

Negative numbers to the power of zero equal minus one.
Zero to the power of zero does not give answer at all (the answer is "indeterminate")


But back to your question.

The meaning of exponentiation could be written as:

3^5 = 1 * 3 * 3 * 3 * 3 * 3

We could use no other number instead of "1" there, so that the answer would be correct.

If you go lower with the powers, one by one, each time you remove one "3", and you still get the correct answer.

Finally, if there are no "3"-s left, we have power of zero, and only single "1" left. So it is only reasonable to use "1" as answer for that. And it works for all the numbers, not just "3".

Answer 4

we know the rule: a^x * a^y = a^(x+y)

any doubt on that?

ok, now put x=0; we get a^0 * a^y = a^y which is correct only if a^0 = 1

what if a=0?

this is a special case... 0^0=0 and a^0=1 where a not equals 0 (a can be + or - )

Answer 5

Well anything BUT a 0 base. The law of exponents allows one to multiply or devide by adding exponents of equal bases. n^1=n, right? n^(-1) = 1/n, right? ...............
n^(1-1)= n^0= n/n = 1 ......... similarly n^(x-x).

Answer 6

any number raised to the power of zero is 1. except zero of course. :P

Answer 7

it is accepted as it is. it comes from the definition this concept.
base can be any real number except zero
if the power zero, the result will be zero

Answer 8

INITIALLY LET US UNDERSTAND EXPONENTS
X^2=X*X
X^3=X*X*X
X^4=X*X*X*X
...............
X^n = X*X*X*X*X*X*X*X........................ n TIMES X

FOR EXAMPLE
2^3 = 2*2*2 = 8
10^2 = 10 * 10 = 100
10^ 5 = 10 * 10 * 10 * 10 * 10 =100000


NOW,

X^(m+n) = X^m * X^n
AND
X^(m-n) = (X^m) / ( X^n)

for example
2^(4 + 3) = (2*2*2*2) * (2*2*2) = (16) * (8) = 128
2^(4 - 3) = (2*2*2*2) / (2*2*2) =2

SO , X^(m-m) = (X^m) / (X^m) "...{ this also means X^0 }...."
X*X*X*X*X*X*X*X..................m TIMES
= _____________________________________
X*X*X*X*X*X*X*X..................m TIMES

= 1

HENCE PROVED

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Answer 9

Yes

Answer 10

x^0 can be written as x^(m-m)
by laws of exponents x^(m-m)=x^m/x^m=1
so anything raaised to power 0 is one