b) \( \left(2 x^{-2}\right)^{-1} \cdot 2 x^{-3} \)

First, let's simplify the expression \( \left(2 x^{-2}\right)^{-1} \). When you take the reciprocal of a term, you flip it over which gives us \( \frac{1}{2 x^{-2}} = \frac{x^{2}}{2} \). Now, substituting back into the expression, we have: \[ \frac{x^{2}}{2} \cdot 2 x^{-3} \] This simplifies to: \[ \frac{2 x^{2} \cdot x^{-3}}{2} \] The '2' cancels out, leaving us with: \[ x^{2} \cdot x^{-3} = x^{2 - 3} = x^{-1} \] So, the final answer is: \[ x^{-1} \] or \(\frac{1}{x}\) if you prefer to express it in a positive exponent form.