Okay so say I have $ (2\cos(x)^{2})^{1/2} $
can anyone show me the proof that $ (2)^{1/2} \cos(x) $ is = to the first one?
http://www.rapidtables.com/math/number/exponent.htm
I was looking at this website and could not find anything relating to different base different exponent.
Your 'equation' $$\sqrt{2(\cos x)^2} = \sqrt{2}\cos x$$ is wrong. Take e.g. $x=\pi,\;$ then the LHS is $\sqrt{2}$ but the RHS evaluates to $-\sqrt{2}.$ Generally you have $\sqrt{a^2}=|a|\;$ for real $a.$