anonymous
  • anonymous
explain why any number (except 0) to the zero power always equals 1
Mathematics
chestercat
  • chestercat
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nikvist
  • nikvist
\[a^0=a^{b-b}=a^b\cdot a^{-b}=a^b\cdot\frac{1}{a^b}=1\]
nowhereman
  • nowhereman
Because 1 is the neutral element of the multiplicative group of real numbers except 0.
anonymous
  • anonymous
please use smaller words "nowhereman" i am going to write this on thetest and i cant sound overlly smart

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anonymous
  • anonymous
nikvist's algebraic interpretation is correct.
nowhereman
  • nowhereman
The above calculation is quite good already. You could also write: \[a^0 \cdot a^n = a^{0+n} = a^n\] so \[a^0 = 1\]
anonymous
  • anonymous
explain why 0 to the 0 power is not one
nowhereman
  • 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)
anonymous
  • 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.
anonymous
  • anonymous
"The choice whether to define 0^0 is based on convenience, not on correctness"
nowhereman
  • nowhereman
Well, what is correctness anyway. After all you choose which axioms you rely on.
anonymous
  • anonymous
Don't shoot the messenger :(
nowhereman
  • nowhereman
Whos quote was it then?
anonymous
  • anonymous
Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999.

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