1. For all real numbers n, m, x, y
Exponent Properties
For all real numbers n, m, x, y
(except that 00 is undefined, and denominators can never be zero)

2. Exponent Properties

3. Exponent Properties
Warning

4. Derivations
We know that
n factors
If n is a positive integer

5. Derivations
Consider
when n > m
For Example:
So adopt the rule:

6. Derivations
Now suppose n = m
But we know that
So it must be true that

7. Derivations
Now suppose n < m, for example:
But our rule says
So we must conclude that

8. Derivations
Or in general,
And by extension