Does ssrt(x) = x^^(1/2)

Notations:
$ssrt(x) = \sqrt {x}_s$  (Super Square Root).
x^^(1/2) $= ^{\frac 12}x$  (half fractional tetrations).

Does $ \sqrt {x}_s =  ^{\frac 12}x$?
Actually no.
because that means
If $a =  ^{b}n$  then $n =  ^{\frac 1b}a$

Let us assume that is true.
if $ ^{n}x = z$ , n aproaches to infinity.
Then $x =  ^{\frac 1n}z=  ^{0}z$
As a fact, we already know that
$ ^{0}z$ is always equal to $1$.
but $x = z^{\frac 1z}$.
Hence the result is $^{0}z = ^{\frac 1z}z$ and that is not true.
Therefore the assumption has failed.