What is the difference between the principal square root of $x$ and $x^{1/2}$? Today I learned that the square root symbol ($\sqrt{}$) represents only the "principal" square root of a number. What about exponents?—if I write $x^{1/2}$, would that encompass the negative square root of $x$ as well?
Edit: $x$ is a non-negative real number.
 A: The idea of "square root" is that it inverts the process of squaring (more generally: $n$th roots invert exponentiation). It is what subtraction is to addition and what division is to multiplication. Exponentiation has another inverse, the logarithm, because it is not a commutative operation, but that's another matter.
If we want to invert squaring, we run into trouble, because we cannot: the function is not injective. $2^2=4$, but also $(-2)^2=4$, so there is no way of retrieving $2$ or $-2$ from $4$ without further information.
This is unlike adding a number, easily reversed, or multiplying a number—reversible if and only if that number is not $0$ (hence, we can't divide by $0$).
Square roots are well behaved: from either square root, $x$, the other square root is $-x$. Thus, we define notation to give a "principal" value, the one that is useful in the most contexts, and it's not hard to find the other value if both are needed (or just the negative one).
Both $\sqrt{x}$ and $x^{1/2}$ are notations to describe the principal square root of $x$. The best notation for the other square root is $-\sqrt{x}$. Example notations that encompass both roots: $\pm \sqrt{x}$ or $\{\sqrt{x},-\sqrt{x}\}$.
As commenters have said, conventions may differ in a context such as complex analysis where it may be important to distinguish between functions (one input, one output) and multifunctions (one input, multiple outputs). It is not universal that, for instance, $x^{1/n}$ is the multifunction and $\sqrt[n]x$ is the function, but some authors may specify this convention and then use it throughout their writing.
If you like, you can use $\sqrt{x}$ to refer to both square roots, but this isn't standard so you need to make it clear before you first use it. Notice that it is then no longer an expression, so values like $\sqrt{4}+1$ also need explaining: does this mean both $-1$ and $3$?
