Prove $\left( \frac{ n }{ n-1 } \right)^{ n-1 } = \left( \frac{ n-1 }{ n } \right)^{ 1-n }$

Question

I have seen this equation:

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

As you can see the numerator switched with the denominator and I wonder how. I know power laws and yet I can not quite figure out what happened here.

Answer 1

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