Question

zero to the power zero equals zero or any number. Please explain in details.

Answer 1

Zero to the power of zero. Zero to the power of zero, denoted by 0^0, is a mathematical expression with no agreed-upon value. The most common possibilities are 1 or leaving the expression undefined.YOU CAN SIMPLY STATE IT TO BE NOT DEFINED THAT IS THE WIDELY ACCEPTED ANSWER.
Below I will give you some postulates which says 0^0 is 1. They have not been proved so just take it as reference.
There are many postulates for 0^0 to be 1 they are

1) The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)^n using the binomial theorem, i.e., 0^n. But the alternating sum of the entries of every row except the top row is 0, since 0^k=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)^0=0^0 if it were defined, should be 1.
2)The limit of x^x as x tends to zero (from the right) is 1. In other words, if we want the x^x function to be right continuous at 0, we should define it to be 1.
3)The expression mn is the product of m with itself n times. Thus m^0, the "empty product", should be 1 (no matter what m is).
4)Another way to view the expression mn is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 0^2=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 0^0=1.
5)Here's an aesthetic reason. A power series is often compactly expressed as
SUMn=0 to INFINITY an (x-c)^n.
We desire this expression to evaluate to a^0 when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a^0 term (not as nice) or by defining 0^0=1.


​​