# Show that the following are equivalent (exponent laws)

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 $$3^{-4}$$).
• @Lil the answer key must contain a mistake: simply substituting $x=0$ yields the statement "$3^{-2}=3^{-4}$", which is false. – Yuval Gat Apr 10 at 16:26