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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? enter image description here

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$(a^b)^c=a^{bc}$, 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 $3^{-4}$).

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  • $\begingroup$ according to the answer key for the worksheet it says these two are equivalent, but I'm struggling to prove this with exponent laws. $\endgroup$ – Lil Apr 10 at 16:25
  • $\begingroup$ @Lil the answer key must contain a mistake: simply substituting $x=0$ yields the statement "$3^{-2}=3^{-4}$", which is false. $\endgroup$ – Yuval Gat Apr 10 at 16:26
  • $\begingroup$ That makes sense. According to this website it says the RHS is a simplification of the LHS $\endgroup$ – Lil Apr 10 at 16:29
  • $\begingroup$ Well, that's incorrect, as i just proved. You can also graph the two in any website like desmos.com and see they're clearly different. $\endgroup$ – Yuval Gat Apr 10 at 16:30

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