1. **State the problem:** Simplify the product of the fractions $$\frac{38}{22} \cdot \frac{66}{9} \cdot \frac{21}{57}$$. 2. **Write the expression:** $$\frac{38}{22} \times \frac{66}{9} \times \frac{21}{57}$$ 3. **Multiply the numerators and denominators:** $$\frac{38 \times 66 \times 21}{22 \times 9 \times 57}$$ 4. **Factor each number to simplify:** - 38 = 2 \times 19 - 22 = 2 \times 11 - 66 = 2 \times 3 \times 11 - 9 = 3 \times 3 - 21 = 3 \times 7 - 57 = 3 \times 19 5. **Rewrite the fraction with factors:** $$\frac{(2 \times 19) \times (2 \times 3 \times 11) \times (3 \times 7)}{(2 \times 11) \times (3 \times 3) \times (3 \times 19)}$$ 6. **Combine all factors:** $$\frac{2 \times 19 \times 2 \times 3 \times 11 \times 3 \times 7}{2 \times 11 \times 3 \times 3 \times 3 \times 19}$$ 7. **Cancel common factors using \cancel{}:** $$\frac{\cancel{2} \times \cancel{19} \times 2 \times \cancel{3} \times \cancel{11} \times 3 \times 7}{\cancel{2} \times \cancel{11} \times \cancel{3} \times 3 \times 3 \times \cancel{19}} = \frac{2 \times 3 \times 7}{3 \times 3}$$ 8. **Simplify numerator and denominator:** $$\frac{2 \times 3 \times 7}{3 \times 3} = \frac{42}{9}$$ 9. **Simplify the fraction $$\frac{42}{9}$$ by dividing numerator and denominator by 3:** $$\frac{\cancel{3} \times 14}{\cancel{3} \times 3} = \frac{14}{3}$$ 10. **Final answer:** $$\boxed{\frac{14}{3}}$$