The meaning of x^(m/n) I guess is the third law.
x^(1/n) means the n-th root of x. If y=x^(1/n) then y^n=x. So y is the n-th root of x.
This is the same as x^(1/n*n)=x^1=x.
y^m=(x^(1/n))^m=x^(1/n*m)=x^(m/n). This is the same as (the n-th root of x) raised to the power m; or it is the n-th root of (x raised to the power m). These mean the same thing. Example: 64^(1/6)=2 so 2^6=64; 64^(2/3)=2^(6*2/3)=2^4=16. 64^(1/3) is the cube root of 64=4. 64^(2/3) is (cube root of 64) squared=4^2=4*4=16. And 64^2=(2^6)^2=2^12, so 64^(2/3)=(64^2)^(1/3)=(2^(6*2))^(1/3)=(2^12)^(1/3)=2^(12/3)=2^4=2*2*2*2=16.