1. The problem is to simplify the expression $$\frac{3^{-2}}{5^{-2}}$$ and verify the equivalence to $$\frac{5^2}{3^2}$$. 2. Recall the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$ for any nonzero number $a$ and positive integer $n$. 3. Apply the negative exponent rule to each term: $$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$ $$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$ 4. Substitute these back into the original expression: $$\frac{3^{-2}}{5^{-2}} = \frac{\frac{1}{9}}{\frac{1}{25}}$$ 5. Dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{\frac{1}{9}}{\frac{1}{25}} = \frac{1}{9} \times \frac{25}{1} = \frac{25}{9}$$ 6. Now, consider the expression $$\frac{5^2}{3^2}$$: $$5^2 = 25$$ $$3^2 = 9$$ 7. So, $$\frac{5^2}{3^2} = \frac{25}{9}$$ 8. Therefore, the original expression simplifies to $$\frac{25}{9}$$, which matches the simplified form of $$\frac{5^2}{3^2}$$. Final answer: $$\frac{3^{-2}}{5^{-2}} = \frac{5^2}{3^2} = \frac{25}{9}$$