Mathematicians define x^0 = 1 for x not equal to zero. Some mathematicians set 0^0 = 1; others say that 0^0 is undefined. The reason for the discrepancy is that in some branches of mathematics (logic, set theory, algebra), the first definition is convenient. In other branches of mathematics (such as analysis - the theory behind calculus), 0^0 is taken as a limit.

By definition, x^1 = x.

Since x^{n + 1} = x(x^n), if we set n = 0, we obtain

x^1 = x(x^0)

Simplifying yields

x = x(x^0)

If x does not equal 0, we may divide each side of the equation by x to obtain

1 = x^0

Any number (aside from 0) to the power of 0 is 1. We can see this quite easily if we look at the example of a number such as 2.

2^1 = 2

2^2 = 4

2^3 = 8

As we can see, every time we raise the power by one, the number doubles. If we then had to lower the power by one, the number would halve.

As such, we can see that:

2^0 = 1

In the case where X is different from 0, X^0 = 1. 0^0 is defined depending on the branch of mathematics you're considering, mainly for convenience.

...X^0 is always equal to 1 except in the case of x=0, where the problems arise. I will not get into this, but is can equal 1,0, or undifined.

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