Question

do these cancel out and why/why not?
S^2 = S^2

S^-2 = S^2

Answer 1

do the same positive power 2 on both sides cancel out
or the opposite - and + powers

Answer 2

Somewhere in your math education, someone has led you astray. "Cancel out" is NOT in math terms, nor does it exist. Perhaps they meant "divide to unity".
example x/x does not cancel out, it divides to unity, so x/x = 1 as long as x is not zero.
Wash your mouth our with soap everytime it speaks "cancel out" .

s^2/s^2 is equal to one as long as s is not zero. if s is a zero, it is a maybe. Wait for the calculus of limits to understand that (example: sinx/x as x approaches zero....x cannot be zero in math rules of division)

s^2/s^-2=s^4.

I hope I "canceled out " your thinking on what "cancel out" means in math: Nothing.