why is x^0 = 1?

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December 5th 2013, 04:24 PM
sakonpure6

why is x^0 = 1?

I do not understand!!! Can some one please explain it to me.
December 5th 2013, 04:28 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.

x^y = e^ln(x)y

x⁰ = e^ln(x)*0 = e⁰ = Exp[0] = 1
December 5th 2013, 04:34 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, 04:40 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, 04:41 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.

how about this then?

x^a = x^(0+a) = x⁰ * x^a

therefore x⁰ = 1

Is that any more digestible?
December 5th 2013, 05:34 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, 06:00 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 x^a = x⁰ * x^a

i.e. z = x^a and c = x⁰

so x⁰ must equal 1
December 5th 2013, 06:19 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 x^a = x⁰ * x^a
i.e. z = x^a 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, 06:37 PM
romsek

Re: why is x^0 = 1?

I didn't want to open up that can of worms.
December 5th 2013, 06:58 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, 10:11 AM
MINOANMAN

Re: why is x^0 = 1?

its only by definition.... no need for ..proofs...they do not mean anything...
December 7th 2013, 10:41 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, 01:16 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, 09:13 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, 02:35 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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