exponentiation - Complicated rational exponents

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.