1. The problem is to simplify the expression $$\left( \frac{3^2}{4^3} \right)^2$$. 2. The formula used here is the power of a quotient rule: $$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$. 3. Apply the rule to the expression: $$\left( \frac{3^2}{4^3} \right)^2 = \frac{(3^2)^2}{(4^3)^2}$$. 4. Simplify the powers inside the numerator and denominator using the power of a power rule: $$ (a^m)^n = a^{m \times n} $$. $$\frac{3^{2 \times 2}}{4^{3 \times 2}} = \frac{3^4}{4^6}$$. 5. Calculate the powers: $$3^4 = 3 \times 3 \times 3 \times 3 = 81$$ $$4^6 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096$$ 6. So the simplified expression is: $$\frac{81}{4096}$$. This fraction cannot be simplified further because 81 and 4096 have no common factors other than 1. Final answer: $$\boxed{\frac{81}{4096}}$$