The unit momentum operator $hat{textbf{p}} ={textbf{p} over |textbf{p}|}$

In studying the Dirac equation, I often come across the unit momentum operator $hat{textbf{p}}$ which is defined as $$hat{textbf{p}} ={textbf{p} over |textbf{p}|},$$where $textbf{p}$ is supposedly the three-momentum vector operator and $|textbf{p}|$ is the magnitude of momentum of the particle. For example, the helicity operator is written as $$h=frac{textbf{S}cdottextbf{p}}{|textbf{p}|}.$$

But $|textbf{p}|$ can also be written in operator form as $$|textbf{p}|=sqrt{hat{p}_x^2+hat{p}_y^2+hat{p}_z^2}.$$

So are we trying to divide the vector operator $textbf{p}$ by another operator? That does not make sense as there is no such thing as operator division. What am I getting wrong?