Sqrt[ab]=Sqrt[a].Sqrt is not defined for imaginary numbers. Which is where this breaks down (your second line). Im sure you already knew that though. Heres one you may have already seen before...spot the breaking of fundamental maths laws...
x^2 - x^2 = x^2 - x^2
(x+x)(x-x)=x(x-x)
x+x=x
2x=x
2=1
1=0
Shown a stupid proof...
[If this is true...my life is ruined