How to promote algebraic expressions to operators in quantum mechanics?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription

$$X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x}$$

in the position basis. Now my questions are:

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  What if $x$ or $p$ appears in the denominator in a classical expression?

  
  
  


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  How to promote this to a quantum expression? What would be the meaning of division by an operator?

  
  

My expression likely contains a mixture of $x$ and $p$. For e.g., it could contain terms like $$\frac{p}{x^2}$$ or $$\frac{xp}{(x^2 + a^2)^{3/2}}.$$

• How to resolve products of non-commuting operators like $x$, $p$ in a satisfactory way?