1. **State the problem:** Calculate the value of the expression $\frac{4}{3} \times 3.14 \times 3^3$. 2. **Recall the formula:** This expression resembles the formula for the volume of a sphere, $V = \frac{4}{3} \pi r^3$, where $\pi \approx 3.14$ and $r = 3$. 3. **Calculate the cube:** Compute $3^3 = 3 \times 3 \times 3 = 27$. 4. **Substitute values:** The expression becomes $\frac{4}{3} \times 3.14 \times 27$. 5. **Multiply numerator terms:** Calculate $4 \times 3.14 \times 27 = 4 \times 84.78 = 339.12$. 6. **Divide by denominator:** Now divide by 3: $$\frac{339.12}{3}$$. 7. **Simplify division:** Show cancellation: $$\frac{\cancel{339.12}}{\cancel{3}} = 113.04$$. 8. **Final answer:** The value of the expression is $113.04$.