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Math Help - Powers of 0 and 1

  1. #1
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    Powers of 0 and 1

    We all know that any number or variable raised to the zero power equals 1
    and that any number or variable raised to the 1st power equals the number or variable.

    SAMPLE A:

    x^0 = 1 as well as 100^0 = 1

    SAMPLE B:

    x^1 = x as well as 100^1 = 100

    Why is this the case?

    Can anyone explain, in simple terms, the WHY concerning the above?
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  2. #2
    MHF Contributor arbolis's Avatar
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    I'll do the why of
    x^0 = 1
    . We have x^0=x^{1-1}=x^1\cdot x^{-1}=\frac{x^1}{x^1}=1. EDIT : Or even better, x^0=e^{0\cdot\ln x}=e^0=exp(0)=1 because \ln(1)=0 (From the definition of the logarithm function. If you define the exponential function as the inverse function of the logarithm one, then what I wrote holds.). Note that it holds \forall x \neq 0.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by magentarita View Post
    We all know that any number or variable raised to the zero power equals 1
    and that any number or variable raised to the 1st power equals the number or variable.

    SAMPLE A:

    x^0 = 1 as well as 100^0 = 1

    SAMPLE B:

    x^1 = x as well as 100^1 = 100

    Why is this the case?

    Can anyone explain, in simple terms, the WHY concerning the above?
    Assuming you buy Arbolis's post then

    x^1=\frac{x^{a+1}}{x^a}

    Or more obvious

    \frac{x^2}{x}

    Assuming x\ne{0}
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  4. #4
    MHF Contributor
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    Thanks A Million!

    I thank you both for that great mathematical insight.
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