1. The problem is to simplify the expression $\left(9^2\right)^{\frac{1}{3}}$ using the properties of rational exponents. 2. Recall the power of a power property: $\left(a^m\right)^n = a^{m \cdot n}$. 3. Apply this property to the expression: $$\left(9^2\right)^{\frac{1}{3}} = 9^{2 \cdot \frac{1}{3}} = 9^{\frac{2}{3}}$$ 4. Therefore, the simplified form of $\left(9^2\right)^{\frac{1}{3}}$ is $9^{\frac{2}{3}}$. 5. Among the given options, $9^{\frac{2}{3}}$ is the correct answer.