I was doing a little bit of fiddling around with complex numbers when I came across this:

$x=1$

$x=sqrt{1}$

$x=sqrt{-1 times -1}$

$x=sqrt{-1} times sqrt{-1}$

$x=itimes i$

$x=i^2$

$1=i^2=-1$

from that, we get that $1=-1$ which, of course, doesn’t make sense. Do the square root laws not work for factorizing $1$ with negative $1$s? Do complex numbers not have the same properties as real numbers? What is the mistake I’ve made?