If x is a non-zero number, what is the value of x^0?

The value of ( x^0 ) for any non-zero number ( x ) is defined as 1. This is based on the properties of exponents. According to these properties, for any non-zero base ( x ), when you raise it to the power of zero, the result is always 1.

This can be understood through the notion of division of powers. For example, if you take ( x^n ) and divide it by ( x^n ) (where ( n ) is a positive integer), the result simplifies to:

[

\frac{x^n}{x^n} = x^{n-n} = x^0

]

Since ( \frac{x^n}{x^n} ) equals 1 (as long as ( x ) is not zero), it follows that ( x^0 ) must equal 1.

Thus, when ( x ) is any non-zero number, raising it to the power of zero yields a consistent and well-established result: 1. This rule is foundational in algebra and is applicable to all non-zero bases.