1. **State the problem:** Simplify the expression $1 \left(\frac{1}{5}\right) m^{4-n} \cdot 0.5 m^{n+6}$. 2. **Rewrite the expression:** $$1 \times \frac{1}{5} \times m^{4-n} \times 0.5 \times m^{n+6}$$ 3. **Multiply the constants:** $$1 \times \frac{1}{5} \times 0.5 = \frac{1}{5} \times 0.5 = \frac{1}{5} \times \frac{1}{2} = \frac{1}{10}$$ 4. **Multiply the powers of $m$ using the rule $m^a \cdot m^b = m^{a+b}$:** $$m^{4-n} \cdot m^{n+6} = m^{(4-n)+(n+6)} = m^{4 - n + n + 6} = m^{10}$$ 5. **Combine the constants and powers:** $$\frac{1}{10} m^{10}$$ 6. **Final simplified expression:** $$\boxed{\frac{1}{10} m^{10}}$$