1. **State the problem:** We need to multiply the mixed number $3 \frac{4}{5}$ by 10 and simplify the result. 2. **Convert the mixed number to an improper fraction:** $$3 \frac{4}{5} = \frac{3 \times 5 + 4}{5} = \frac{15 + 4}{5} = \frac{19}{5}$$ 3. **Multiply the improper fraction by 10:** $$\frac{19}{5} \times 10 = \frac{19 \times 10}{5} = \frac{190}{5}$$ 4. **Simplify the fraction:** $$\frac{190}{5} = 38$$ 5. **Final answer:** The product of $3 \frac{4}{5}$ and 10 in simplest form is $38$.