1. **Simplify** $\sqrt{22m^7} \cdot \sqrt{8m^2}$. We use the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. $$\sqrt{22m^7} \cdot \sqrt{8m^2} = \sqrt{22m^7 \cdot 8m^2} = \sqrt{176m^9}$$ 2. **Simplify** $\sqrt{176m^9}$. Factor 176 as $16 \times 11$: $$\sqrt{176m^9} = \sqrt{16 \times 11 \times m^9} = \sqrt{16} \cdot \sqrt{11} \cdot \sqrt{m^9}$$ Since $\sqrt{16} = 4$ and $\sqrt{m^9} = m^{\frac{9}{2}} = m^4 \cdot \sqrt{m}$: $$4 \cdot \sqrt{11} \cdot m^4 \cdot \sqrt{m} = 4m^4 \sqrt{11m}$$ **Final answer:** $4m^4 \sqrt{11m}$