What is Zero to the Power of Zero equal to?

First of all, you need to understand why any number raised to the Power of Zero equals 1. To explain this, I'm providing you an Example.

Example No. 1: Prove that   

We know that: 

From L.H.S:

Now, From (i) and (ii) :

Hence, the statement is Proved.

In General, we can write:

Where " x " can be any number except:

Because, if we take " x = 0 " , then we will be get:

 

Which is not equal to 1. This expression is considered Indeterminate rather than having a definitive value. 

Basically, division is the inverse of Multiplication. So, for any number a and b (where b ≠ 0):

if and only if

Now, let's apply this idea to: 

We want to find "c" such that:

This equation is true for any value of "c", since multiplying 0 by any number always results in zero. Therefore, the equation holds for all values of c, implying that 0/0 has Infinite Possible Solutions. In Calculus and Algebra, this expression 0/0 is called an Indeterminate Form. Hence, it's not assigned a specific value in standard Arithmetic and is considered undefined. 

In some contexts, you might encounter 0/0 being treated as equal to 1, particularly in Combinatorics and Computer Programming Languages. In combinatorics, 0/0 is often defined as 1 for convenience, especially when calculating binomial coefficients. 

For Example, in the formula for choosing Zero items from Zero possibilities, it makes sense to define;

To maintain consistency with the general formula for Binomial Coefficients. Similarly, in many Programming Languages, defining;

can simplify Algorithms, especially in scenarios involving empty products or when computing powers. However, it's important to remember that while these definitions are useful in specific fields, and;

is still considered as Indeterminate in nature in most of the Mathematical contexts.

Bonus:

The expression;

is called undefined because division by zero is not meaningful within standard Arithmetic and Mathematics. We know that, division is defined as the inverse of multiplication.

For a division operation:

It must hold that:

Now, consider;

Substituting into the multiplication equation:

But multiplying 0 by any number always results in 0:

It is impossible for 0 multiply by c to equal 1. Therefore, the division (1/0) does not satisfy the rules of Arithmetic, and we say it is undefined.

While we sometimes write it as equal to infinity, 

This is only a symbolic representation used in certain contexts, such as Limits in Calculus. It is important to note that infinity is not a number but rather a concept used to describe unbounded behavior, and division by zero remains undefined in strict mathematics terms.