I was asked recently why . Remember that if n is a positive whole number then . Clearly you can’t multiply 2 by itself 0 times 😕
The key, when extending properties of the number system, is to use definitions that work for every number. So, for example
which gives you the rule that
- to divide powers you subtract the indices (the small superscripted numbers)
This leads to
Similarly, for any positive real number.
And if the number is negative? Great care is needed in this case. For example, using only real numbers, but is not a real number. The problem arises because the general definition of a power is given by and is undefined if a is negative or 0. Using complex numbers (which helps with ) just makes things more complex 😕 – see Log of Complex Number