1. Stating the problem: Solve the equation $3^{x-2} = \left(\frac{1}{3}\right)^{-2x}$ for $x$. 2. Recall the property of exponents: $\left(\frac{1}{a}\right)^b = a^{-b}$. 3. Rewrite the right side using this property: $$\left(\frac{1}{3}\right)^{-2x} = 3^{2x}$$ 4. Now the equation is: $$3^{x-2} = 3^{2x}$$ 5. Since the bases are the same and nonzero, set the exponents equal: $$x - 2 = 2x$$ 6. Solve for $x$: $$x - 2 = 2x$$ $$x - \cancel{x} - 2 = 2x - \cancel{x}$$ $$-2 = x$$ 7. Final answer: $$x = -2$$