integer - What does fractional power of number mean -


I have a question in my mind for many years. I know that it looks stupid and if people do not like it then they will be removed.

I know that 2 ^ 3 means that 2 to 3 times times, 3 ^ -2 means multiplication (1/3) twice. 2 ^ (0.22) times 2 times? I mean what is the logic behind partial power of integer? 

  a ^ (m / n) = n√ (a ^ m)

This is a ^ MKN-J root 

  0.22 = 22/100m = 22n = 100

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