1. **State the problem:** Simplify the expression $$\sqrt{144m^{16}n^{20}}$$. 2. **Recall the formula and rules:** The square root of a product is the product of the square roots: $$\sqrt{a b} = \sqrt{a} \times \sqrt{b}$$. Also, for even powers, $$\sqrt{x^{2k}} = x^k$$ if $$x \geq 0$$. 3. **Apply the square root to each factor:** $$\sqrt{144m^{16}n^{20}} = \sqrt{144} \times \sqrt{m^{16}} \times \sqrt{n^{20}}$$ 4. **Simplify each square root:** - $$\sqrt{144} = 12$$ because $$12^2 = 144$$. - $$\sqrt{m^{16}} = m^{\frac{16}{2}} = m^8$$. - $$\sqrt{n^{20}} = n^{\frac{20}{2}} = n^{10}$$. 5. **Combine the simplified parts:** $$12 \times m^8 \times n^{10} = 12m^8n^{10}$$ 6. **Final answer:** $$\boxed{12m^8n^{10}}$$