Do $a^{1/n}$ and $n$th root of a mean the same thing? I encountered a question in which I had to find $x$ and the question had $3^{1/x}$ and I got $x=1/2$ and $1/4$ as its solution. But my textbook said that "$x$th root of $3$" is valid for $x\geq 2$ and $x$ should be a natural number.
But the question did not directly mention $x$th root 3 and instead "$3^{1/x}$". Do they actually mean the same? Do they have same restrictions on $x$?
PS: The actual question was "Solve for real values of $x$: log $4$ + ($1$+ $1/2x$) log $3$= log ($3^{1/x}$ + $27$)"

Well, in general they mean the same.
but it mostly depends on who is asking you the question, so they might have one of the following differences:

*

*$\root 2 \of 4$ means both $-1$ and $1$ while $4^{\frac{1}{2}}$ means just the principle root, $2$.

*it might hint that negative numbers are to be used as exponents as the expression $\root -1 \of {2} = \frac{1}{2}$ is practically never used.

*alot less popular but some writers define $\root 0 \of n = \begin{cases}
1,  & n>0 \\
\text{undefined}, & \text{otherwise}
\end{cases}$
The best option is to ask the professor or lecturer.

