1. **State the problem:** Simplify the expression $$\frac{4(m^4)^3}{5m^5}$$ using only positive exponents. 2. **Apply the power of a power rule:** When raising a power to another power, multiply the exponents. $$ (m^4)^3 = m^{4 \times 3} = m^{12} $$ 3. **Rewrite the expression:** $$ \frac{4m^{12}}{5m^5} $$ 4. **Divide powers with the same base:** Subtract the exponents. $$ \frac{4\cancel{m^{12}}}{5\cancel{m^5}} = \frac{4}{5} m^{12-5} = \frac{4}{5} m^7 $$ 5. **Final answer:** $$ \frac{4}{5} m^7 $$ This is the simplest form with only positive exponents.