Question: How is this $\frac{m}{n}$ equals to $m \times \frac{1}{n}$? Any logical proof for this? Which draws this conclusion:

> 
> A fractional exponent like $\frac{m}{n}$ means do the $m^{\text{th}}$ power, then take the $n^{\text{th}}$ root or take the $n^{\text{th}}$ root and then do the $m^{\text{th}}$ power.
> 
> 
>

Answer: 
It is the same; you just change the order of the operations but it does not affect the result. Using logarithms would show it in a simple way.
If you are making a number bigger by multiplying by $m$, then making it smaller by dividing it by $n$, it will be the same as making it smaller by $n$ then making it bigger by $m$. I guess it's just how numbers work.