## anonymous 5 years ago explain why any number (except 0) to the zero power always equals 1

1. anonymous

$a^0=a^{b-b}=a^b\cdot a^{-b}=a^b\cdot\frac{1}{a^b}=1$

2. nowhereman

Because 1 is the neutral element of the multiplicative group of real numbers except 0.

3. anonymous

please use smaller words "nowhereman" i am going to write this on thetest and i cant sound overlly smart

4. anonymous

nikvist's algebraic interpretation is correct.

5. nowhereman

The above calculation is quite good already. You could also write: $a^0 \cdot a^n = a^{0+n} = a^n$ so $a^0 = 1$

6. anonymous

explain why 0 to the 0 power is not one

7. nowhereman

Because it is undefined. For the exponential function 0^x to be continuous it must be 1 but for x^0 to be continuous it must be 0. So you can't define it consistently (e.g. so that all power-rules hold)

8. anonymous

Or you can go back to nikvist's explanation and realize that to get $0^0$ you'd have to divide by 0 which isn't defined.

9. anonymous

"The choice whether to define 0^0 is based on convenience, not on correctness"

10. nowhereman

Well, what is correctness anyway. After all you choose which axioms you rely on.

11. anonymous

Don't shoot the messenger :(

12. nowhereman

Whos quote was it then?

13. anonymous

Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999.