Is $x^{1/2}$ equal to a positive value only?

Question

My friend argues that $\sqrt{x}$ is unequal to $x^{1/2}$ because $\sqrt{x}$ can equal both negative and positive values, whilst $x^{1/2}$ can equal postive values only.

I tried to research it, but haven't found enough on that matter and I think that $x^{1/2}$ is simply only another way of representing $\sqrt{x}$. Which one of us is right?

Answer 1

First of all :

$$x^{1/2}=\sqrt{x}$$

Also, $\sqrt{x} ~(= x^{1/2})$ is always positive.

Note : $y^2=x$ and $y= \sqrt{x}$ are different.

The first one allows $y =\pm \sqrt{x}$ whereas the second one only refers to $y =+\sqrt x$

Answer 2

The convention for positive real numbers is that $x^{1/2} \equiv \sqrt{x}$ denotes the positive square root of $x$, hence why, for example, the quadratic formula has the $\pm\sqrt{b^2-4ac}$ in the numerator. The same is true for $x^{\alpha}$ for $\alpha$ not an integer (so this includes $1/2,1/3,\pi, \log{2}$ and so on).