# why is x^0 = 1?

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• December 5th 2013, 03:24 PM
sakonpure6
why is x^0 = 1?
I do not understand!!! Can some one please explain it to me.
• December 5th 2013, 03:28 PM
romsek
Re: why is x^0 = 1?
Quote:

Originally Posted by sakonpure6
I do not understand!!! Can some one please explain it to me.

xy = eln(x)y

x0 = eln(x)*0 = e0 = Exp[0] = 1
• December 5th 2013, 03:34 PM
sakonpure6
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?
• December 5th 2013, 03:40 PM
Plato
Re: why is x^0 = 1?
Quote:

Originally Posted by sakonpure6
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 $1=\frac{x^1}{x^1}=x^{1-1}=x^0$
• December 5th 2013, 03:41 PM
romsek
Re: why is x^0 = 1?
Quote:

Originally Posted by sakonpure6
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.

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

therefore x0 = 1

Is that any more digestible?
• December 5th 2013, 04:34 PM
sakonpure6
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 :)
• December 5th 2013, 05:00 PM
romsek
Re: why is x^0 = 1?
Quote:

Originally Posted by sakonpure6
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
• December 5th 2013, 05:19 PM
Plato
Re: why is x^0 = 1?
Quote:

Originally Posted by romsek
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.
• December 5th 2013, 05:37 PM
romsek
Re: why is x^0 = 1?
I didn't want to open up that can of worms.
• December 5th 2013, 05:58 PM
Prove It
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*}.
• December 7th 2013, 09:11 AM
MINOANMAN
Re: why is x^0 = 1?
its only by definition.... no need for ..proofs...they do not mean anything...
• December 7th 2013, 09:41 AM
Plato
Re: why is x^0 = 1?
Quote:

Originally Posted by MINOANMAN
its only by definition.... no need for ..proofs...they do not mean anything...

That is simply not the case.
• December 7th 2013, 12:16 PM
Hartlw
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.
• December 7th 2013, 08:13 PM
MINOANMAN
Re: why is x^0 = 1?
Quote:

Originally Posted by Plato
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..
• December 8th 2013, 01:35 AM
topsquark
Re: why is x^0 = 1?
Quote:

Originally Posted by MINOANMAN
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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