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Math Help - why is x^0 = 1?

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    Senior Member sakonpure6's Avatar
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    why is x^0 = 1?

    I do not understand!!! Can some one please explain it to me.
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    Re: why is x^0 = 1?

    Quote Originally Posted by sakonpure6 View Post
    I do not understand!!! Can some one please explain it to me.
    xy = eln(x)y

    x0 = eln(x)*0 = e0 = Exp[0] = 1
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    Senior Member sakonpure6's Avatar
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    Re: why is x^0 = 1?

    I am sorry, but this is a bit too advanced for me, mind explaining what e ^ ln(x)y means?
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    Re: why is x^0 = 1?

    Quote Originally Posted by sakonpure6 View Post
    I do not understand!!! Can some one please explain it to me.
    Here is a less obscure reason.
    First of all, there is some debate about what 0^0 could mean.

    So let's say that x\ne 0~.
    Then \frac{x}{x}=1 but that means \frac{x^1}{x^1}=1.

    So by the laws of exponents we have 1=\frac{x^1}{x^1}=x^{1-1}=x^0
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    Re: why is x^0 = 1?

    Quote Originally Posted by sakonpure6 View Post
    I am sorry, but this is a bit too advanced for me, mind explaining what e ^ ln(x)y means?
    hmm, I'm wondering how to show this w/o using log and exponential functions then.

    how about this then?

    xa = x(0+a) = x0 * xa

    therefore x0 = 1

    Is that any more digestible?
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    Senior Member sakonpure6's Avatar
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    Re: why is x^0 = 1?

    How did we conclude x^0 dont we generaly add the exponents when multiplying them? And how did you conclude the1? I really appreciate it.

    @Plato, Ooohh, I see. Thanks
    Last edited by sakonpure6; December 5th 2013 at 04:58 PM.
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    Re: why is x^0 = 1?

    Quote Originally Posted by sakonpure6 View Post
    How did we conclude x^0 dont we generaly add the exponents when multiplying them? And how did you conclude the1? I really appreciate it.
    if z = c * z then c=1 as 1 is the multiplicative identity on the real numbers. ok?

    we have xa = x0 * xa

    i.e. z = xa and c = x0

    so x0 must equal 1
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    Re: why is x^0 = 1?

    Quote Originally Posted by romsek View Post
    if z = c * z then c=1 as 1 is the multiplicative identity on the real numbers. ok?
    we have xa = x0 * xa
    i.e. z = xa and c = x0
    so x0 must equal 1
    Only if x\ne 0.

    One can see here how complicated this question has become.
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    Re: why is x^0 = 1?

    I didn't want to open up that can of worms.
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    Re: why is x^0 = 1?

    From the index law \displaystyle \begin{align*} \frac{a^m}{a^n} = a^{m - n} \end{align*}, in the case where \displaystyle \begin{align*} m = n \end{align*} we have

    \displaystyle \begin{align*} LHS &= \frac{a^m}{a^n} \\ &= \frac{a^m}{a^m} \\ &= 1 \end{align*}

    but we also have

    \displaystyle \begin{align*} RHS &= a^{m - n} \\ &= a^{m - m} \\ &= a^0 \end{align*}

    Thus \displaystyle \begin{align*} a^0 = 1 \end{align*}.

    This is of course provided that \displaystyle \begin{align*} a \neq 0 \end{align*}.
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    Re: why is x^0 = 1?

    its only by definition.... no need for ..proofs...they do not mean anything...
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    Re: why is x^0 = 1?

    Quote Originally Posted by MINOANMAN View Post
    its only by definition.... no need for ..proofs...they do not mean anything...
    That is simply not the case.
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    Re: why is x^0 = 1?

    Whoops. Too fast. Already done by Plato and Romsek (integer exponents). Missed it. Oh well, I'm not the only one.
    My apologies.
    Last edited by Hartlw; December 7th 2013 at 12:23 PM.
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    Re: why is x^0 = 1?

    Quote Originally Posted by Plato View Post
    That is simply not the case.
    yes it is the case mathematics is not just computations...it is philosophy..x^0 is meaningless in mathematics unless you define it...you cannot just supply "proofs"...x^0 is by definition =1 as it is 0! =1 .... No Further comments..
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    Re: why is x^0 = 1?

    Quote Originally Posted by MINOANMAN View Post
    yes it is the case mathematics is not just computations...it is philosophy..x^0 is meaningless in mathematics unless you define it...you cannot just supply "proofs"...x^0 is by definition =1 as it is 0! =1 .... No Further comments..
    The debate as to whether 0^0 is equal to 1 or 0 is philosophy. x^0 = 1 (for non-zero x) is not philosophy...It can be unambiguously derived from the laws of exponents, as has been shown in several posts in this thread.

    -Dan
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