1. **State the problem:** Calculate the value of the expression $\frac{1}{3} \times 3.14 \times 3^2 \times 7$. 2. **Recall the order of operations:** Multiply and exponentiate from left to right. 3. **Calculate the exponent:** $$3^2 = 9$$ 4. **Substitute back into the expression:** $$\frac{1}{3} \times 3.14 \times 9 \times 7$$ 5. **Multiply the constants first:** $$3.14 \times 9 = 28.26$$ 6. **Now multiply by 7:** $$28.26 \times 7 = 197.82$$ 7. **Finally, multiply by $\frac{1}{3}$:** $$\frac{1}{3} \times 197.82 = \frac{\cancel{1}}{\cancel{3}} \times 197.82 = \frac{197.82}{3} = 65.94$$ **Final answer:** $$65.94$$