Question about Properties of Exponents

Question

I'm just doing some assignments and I was a little confused by this, could someone explain why the second equation is not equal like the first one? From what I've learned, if you take the inverse you switch the exponential sign (positive to neg for example) $$\frac{3^{1}} {1^{}} = \frac{1^{}} {3^{-1}}$$ $$\frac{1^{1}} {3^{}} ≠ \frac{3^{}} {1^{-1}}$$

Answer 1

What is true is $$\left( \frac{a}{b} \right)^{-1} = \frac{b}{a}$$

For the first equation.

$$\frac{1}{3^{-1}}=\frac{1^{-1}}{3^{-1}}=\left( \frac13\right)^{-1}=\frac{3}{1}$$

but for the second equation

$$\frac{3}{1^{-1}} \neq \left(\frac31 \right)^{-1}$$