# 5.4 Exponents with a Power of 0

For all x≠0, x^0=1

We're now ready to understand what x0 could possibly mean. We just developed some rules for counting numbers and we'd like them to hold for any possible integer; so if our rules hold for 0 as well, then:

Explore!

 3^2 \times 3^0=3
 5^4\times 5^0= 5

In each case, if we multiply an exponential by this mysterious exponential with a 0 as the exponent, we get back exactly what we started with.

But, this must mean that 30 and 50 are both equal to .

In general, then, it would seem that we have that for any x value:

x^0 =1

It turns out that we don't actually define 00; the explanation as to why we don't allow this would best be understood if you understood logarithms, which is something that you'll study in pre-calculus.

Remember our original model? We thought that our formula didn't work for n=0. But, now we see that it DOES work. Using the model that we developed for counting numbers,

P(n)=1000(1.02)^n,

we see that:

P(0) = 1000(1.02)^0= 1000 \cdot 1 = 1000

which matches our table exactly! Now that we understand what x0 means, we can begin thinking about negative exponents.