One of the great things about mathematics is that its rules build upon one another, using simple mathematical operations to prove more complex mathematical truths. Raising a number to the power of zero is no exception — you can prove that

*n*^{0} = 1 by relying on simpler mathematical properties that you already know.

In this case, the two properties you need to know are

*n*^{x} × *n*^{y} = *n*^{x}^{+y}
- The
*associative* property of multiplication: (*xy*)*z* = *x*(*yz*)

Equation (a) is easy enough to show simply by choosing a couple of exponents and writing out the entire equation *without* using exponents, like this:

*n*^{3} × *n*^{4} = (*n* × *n* × *n*) × (*n* × *n* × *n* × *n*)

Because of the associative property of multiplication [see (b) above], you know that you can eliminate the parentheses and arrive at this:

*n*^{3} × *n*^{4} = *n* × *n* × *n* × *n* × *n* × *n* × *n* = *n*^{7}

No matter what numbers or what exponents you try (unless you use zero as the base number), *n*^{x} × *n*^{y} = *n*^{x}^{+y} always.

With these two simple properties, you can better understand how raising to the power of zero works. Using what you've learned above, solve this equation:

*n*^{4} × *n*^{0} = ???

Because of (a) above, you know that

*n*^{4} × *n*^{0} = *n*^{4+0} = *n*^{4}

The only way that *n*^{4} × *n*^{0} =* n*^{4} is if *n*^{0} = 1. Plugging real, non-zero numbers into an equation like this will yield the same results.

If you understand how negative exponents work, you could also take a different route to prove that *n*^{0} = 1. (*Hint:* *n*^{–x} = 1/*n*^{x}) Pick any non-zero number for *n* and solve this equation:

*n*^{–5} × n^{5} = ???

I'll leave it to you to figure it out.