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anonymous

  • 5 years ago

explain why any number (except 0) to the zero power always equals 1

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  1. nikvist
    • 5 years ago
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    \[a^0=a^{b-b}=a^b\cdot a^{-b}=a^b\cdot\frac{1}{a^b}=1\]

  2. nowhereman
    • 5 years ago
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    Because 1 is the neutral element of the multiplicative group of real numbers except 0.

  3. anonymous
    • 5 years ago
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    please use smaller words "nowhereman" i am going to write this on thetest and i cant sound overlly smart

  4. anonymous
    • 5 years ago
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    nikvist's algebraic interpretation is correct.

  5. nowhereman
    • 5 years ago
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    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
    • 5 years ago
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    explain why 0 to the 0 power is not one

  7. nowhereman
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    "The choice whether to define 0^0 is based on convenience, not on correctness"

  10. nowhereman
    • 5 years ago
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    Well, what is correctness anyway. After all you choose which axioms you rely on.

  11. anonymous
    • 5 years ago
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    Don't shoot the messenger :(

  12. nowhereman
    • 5 years ago
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    Whos quote was it then?

  13. anonymous
    • 5 years ago
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    Donald C. Benson, The Moment of Proof : Mathematical Epiphanies. New York Oxford University Press (UK), 1999.

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