Thread: To the power of zero; zero exponent

1. To the power of zero; zero exponent

I searched for a discussion of the rules governing exponents of zero but didn't find anything so I thought I'd start a thread on it.

We have learned that any number raised to the power of zero is equal to 1. Why is that so? Also, does this apply to zero, that is, zero raised to the power of zero?

2. To answer this question is beating a dead horse. But it is summer.
We know the rules of exponents: $\frac{{x^5 }}{{x^3 }} = x^{5 - 3} = x^2$.
Therefore we would have to allow $1 = \frac{{x^5 }}{{x^5 }} = x^{5 - 5} = x^0$.
But we never allow division by zero!
Therefore, it can be argued that $0^0$ is not defined.
But if you look at the first link, you will see that this is very old argument.
The great Euler argued that $0^0 = 1$.

Math Forum: Ask Dr. Math FAQ: Zero to Zero Power
Math Forum: Ask Dr. Math FAQ: N to Zero Power

3. I have heard of conventions stating that $0^0 = 1$ and $0^0 = 0$. But these are just conventions used in specific cases to generalize a result. As Plato says, this is not a defined operation so it is undefined.

-Dan

4. Originally Posted by Cifrocco
We have learned that any number raised to the power of zero is equal to 1. Why is that so? Also, does this apply to zero, that is, zero raised to the power of zero?
(e^x) (e^0)= e^x+0 = e^x and hence e^0 = 1 for all e in the set R (Real numbers)

What is 0^0?? I asked my Maths Professor that question 15 years ago; His answer was: God knows

5. Originally Posted by Math's-only-a-game
(e^x) (e^0)= e^x+0 = e^x and hence e^0 = 1 for all e in the set R (Real numbers)
For all? Did you then divide by zero?

6. Originally Posted by Plato
For all? Did you then divide by zero?
Are you trying to say: If for all e in R then e = 0 ==> e^x = 0^0 = ??

Well spotted; I should have said "for all {R - 0}" i.e for all Real numbers except 0.

My Apologies

Btw I have sent you a private message; have you received it?? I was asking how do you type/copy/paste the maths text. I have try it with Mathcad, Math type 6 and Word with no success.

I would highly appreciate any help from anyone.

7. Originally Posted by Math's-only-a-game
[COLOR=navy]Btw I have sent you a private message; have you received it?? I was asking how do you type/copy/paste the maths text. I have try it with Mathcad, Math type 6 and Word with no success.
Using MathType you can produce LaTeX code. Choose LaTeX tranlators found under Preferences. By coping an expression you get this code $\frac{{\partial x}}{{\partial y}}$ which you then edit to $$\frac{{\partial x}}{{\partial y}}$$ to produce $
\frac{{\partial x}}{{\partial y}}$
.

8. Originally Posted by Plato
Using MathType you can produce LaTeX code. Choose LaTeX tranlators found under Preferences. By coping an expression you get this code $\frac{{\partial x}}{{\partial y}}$ which you then edit to $$\frac{{\partial x}}{{\partial y}}$$ to produce $
\frac{{\partial x}}{{\partial y}}$
.
Thanks for the info. However I can't get the preferences options activated possibly because I have the free version. In other words to use the preferences options I would have to buy the software.

Thanks anyway. Much obliged

9. Originally Posted by Math's-only-a-game
I can't get the preferences options activated possibly because I have the free version. In other words to use the preferences options I would have to buy the software.
Even with the free version.,TeXAid, it works. Go to perferences, pulldown that tab the first option is Translators. Ckick translators an new window appears. Click translate to other languages(text). In the window below that select TeX--LaTex2.09 and later. Uncheck any boxes below that.

10. Originally Posted by Plato
Even with the free version.,TeXAid, it works. Go to perferences, pulldown that tab the first option is Translators. Ckick translators an new window appears. Click translate to other languages(text). In the window below that select TeX--LaTex2.09 and later. Uncheck any boxes below that.
That's exactly what I've tried but for some reason none of the options under preferences can be activated. Maybe the software is corrupted. I should try to re-download the software but I don't know if they will allow me to do so, for I used my free trial.

Zack

11. Download TeXAide. It is free. But does not work with Word.

12. Originally Posted by Plato
${\rm x}^{\rm n} {\rm + y}^{\rm n} \ne {\rm z}^{\rm n} {\rm }\forall {\rm n > 2}$

Yes, it worked!!!

I tried to leave space between the ${\rm z}^{\rm n}$ the "for all" and n but no success.

No matter; it's still much better than the rubbish I was posting before.

Thanks Plato; you are a good man .

13. Originally Posted by Plato
We know the rules of exponents: $\frac{{x^5 }}{{x^3 }} = x^{5 - 3} = x^2$.
Therefore we would have to allow $1 = \frac{{x^5 }}{{x^5 }} = x^{5 - 5} = x^0$.
But we never allow division by zero!
Stupid question... Why does the argument not apply to all powers of zero? ( $0^n=0^{n+1}/0^1=0/0$)

14. Originally Posted by sleepingcat
Stupid question... Why does the argument not apply to all powers of zero? ( $0^n=0^{n+1}/0^1=0/0$)
dividing by zero is invalid, always. 0^n would be defined otherwise. for instance, if n is an integer, we may define it as 0 times itself n times, and so on

15. I would like to add, that although straight up $0^0$ is undefined

Things of the form $\lim_{x\to{c}}f(x)^{g(x)}$ where $f(c)=0$ and $g(c)=0$ can yield theoretically and value one chooses

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