# Mathematics Weblog

## Powers

Friday 25 June 2004 at 3:37 pm | In Articles | 3 CommentsI was asked recently why . Remember that if *n* is a positive whole number then . Clearly you can’t multiply 2 by itself 0 times ðŸ˜•

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.

What about zero powers of non-negative powers? is a controversial case I have mentioned on 29 February (Q2.) and see Dr Math FAQ for more on this.

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 ðŸ˜• – see Log of Complex Number

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I didn’t spot this article before… I would say that 1 is the multiplicative identity.

Or, in other words,

we know from laws of indices that:

2^0 * 2^n = 2^(0+n)

This goes for any base with the exception of 0

Even in the case of 0, I would be tempted to say that 0^0 is

more 1 than anything elseComment by Ronald — Wednesday 22 December 2004 11:42 pm #

missing superscripts… sorry

Comment by Ronald — Wednesday 22 December 2004 11:44 pm #

I have added them in for you ðŸ™‚

Comment by Steve — Wednesday 22 December 2004 11:48 pm #