Determine whether the two expressions are equivalent
I was able to solve this by distributing the -2(1+x$^2$) and setting it equal to the expression x$^2$-4 and getting x$^2$=2/3 and then substituting and checking, but how can I confirm the expressions are equivalent this just by simplifying each side of the expression-kind of like a proof? 
$(a^b)^c=a^{bc}$, so $\left(3^{-2}\right)^{1+x^2}=\left(3^{(-2)\cdot (1+x^2)}\right)=3^{-2x^2-2}\ne 3^{x^2-4}$ (for example, equality fails at $x=0$ since then the LHS is $3^{-2}$ while the RHS is $3^{-4}$).