Why is anything to the power 0 = 1?

Question

Any proofs of this?

Answer 1

WE KNOW THAT (a^1/a^1)=1
the next step if you bring the denominator 'a' to the numerator.
it becomes (a^1-1)=1.
Now a^0=1

Answer 2

just think of a fraction with the same numerator and denominator.lets take the following example:

4/4

these will result to 1, right?

the numerator has a power of 1 and so do with the denominator. they also have the same digits.

if we subtract the exponents of the said digits and retain 4,we will have the following answer:

4^0

so 4/4=4^0=1

this is applicable to all numbers except 0 because 0 should not be placed in the denominator because anything divided to 0 is always undefined.

Answer 3

There isn't a proof because it's not a mathematical law. It's actually a convenient definition. The real answer is that it's undefined.

We define it to be that way for the sake of continuity. All of the exponentials have that fixed point on the y -axis which corresponds to this law, and it's worked so far.

The reason you can't prove it is because you'd eventually end up taking the logarithm of zero or dividing by zero (That's a guess, by the way), both of which are undefined operations.

The links here provide good practical proof that it should be, but it isn't rigorous.

Answer 4

a^0 = 1

Proof:
(a^n)/(a^n) = a^(n-n)

But left hand side is equal to 1. Thus
the right hand side must be equal to 1 also.

Since a^(n-n) = 1 and a^(n-n) = a^0
It follows that a^0 = 1 by transitivity of
equality.
Q.E.D.

Answer 5

For any non-zero number x,

x/x = 1

Since,

x/x = x^1/x^1

= (x^1)(x^-1)

= x^(1-1)

= x^0

Therefore x^0 = 1 for all non-zero x

Answer 6

anything raised to zero is 1, except zero itself.

In symbols,
a^0 = 1, a is not equal to 0.

Because, 0 = R - R, where R is any real number not equal to zero.
a^0 = a^(R - R)

Since b^(x - y) = b^x/b^y, then
a^0 = a^(R - R) = a^R/a^R

Since a^R/a^R = 1, then
a^0 = a^(R - R) = a^R/a^R = 1

Therefore,
a^0 = 1

Why a cannot be zero?

Since
a^0 = a^(R - R) = a^R/a^R
if a = 0,
0^0 = 0^(R - R) = 0^R/0^R
Since 0^R = 0,
0^0 = 0^(R - R) = 0^R/0^R = 0/0 = indeterminate...

Therefore,
0^0 = indeterminate
QED
^_^

Answer 7

(a ^ 1) / (a ^ 1) = 1
because a number divided by itself is 1... mathematical rule
a ^ (1 -1) = 1
a ^ 0 = 1

Answer 8

for example we have the fraction x/x
we all know that the exponent of x is 1
and when we simplify fractions with variables
we subtract the exponent of the same variables
so 1-1=0
x^0=1

Answer 9

this not alway ture becouse when 0^0 i undefind thing
therefore x<>0

1=x/x

1=x/x = x^1/x^1

1= (x^1)(x^-1)

1= x^(1-1)

1= x^0

Answer 10

x^a * x^b = x^(a+b)
put a = 1 and b = -1

x/x = x^0 =1
naturally x cannot be zero