Domain for index of Radical Sign What is the domain of $\sqrt[x]{a}$, and is $\sqrt[x]{a}=a^{1/x}$ always true??
I was told that the domain of $\sqrt[x]{a}$ is natural numbers and the domain of $a^{1/x}$ is real numbers, so they are not identical. Is it true??

Were you also told to not split infinitives and that sentences shouldn't end with prepositions like "by" or "with"?
By convention we agree that $\root x \of a$ refers to the principal $x$th root of $a$. If $x$ is a positive real integer, then $a$ has precisely $x$ roots, and each of the other roots can be obtained by multiplying the principal root by the appropriate $x$th root of 1.
For example, $\root 4 \of 4$ is the principal root of $x^4 - 4$, which of course can be simplified to $\sqrt 2$, roughly 1.414213562373. The other roots are $-\sqrt 2$, $i \sqrt 2$ and $-i \sqrt 2$, which consist of $\sqrt 2$ multiplied by the quartic roots of 1 in turn: $-1$, $i$, $-i$.
And just to make sure I get some flack for this answer, I'm going to say these four roots can just as validly be expressed as $\sqrt 2$, $-\sqrt 2$, $\sqrt{-2}$ and $-\sqrt{-2}$.
In the case of $\sqrt{-2}$, it has two roots, and we can probably come to the agreement that it represents the principal root of 2 (which is $\sqrt 2$) multiplied by $i$.
Some people will be pedantic and tell you that $\sqrt{-2}$ is undefined and you really should write $i \sqrt 2$. But if everyone understands that's what you mean, then what is the problem?
If we agree that the radical symbol stands for a principal root, we should also agree that $a^{\frac{1}{x}}$ also stands for a principal root.
Lastly, I'd like to mention that, depending on your TeX installation, you may or may not be able to use \root x \of a instead of \sqrt[x]{a}. I personally prefer the former to the latter.
