1. **Problem:** Find the root: $5\sqrt{32a^5 b^{10}}$ 2. **Formula and rules:** The $n$th root of a product is the product of the $n$th roots: $$\sqrt[n]{xy} = \sqrt[n]{x} \times \sqrt[n]{y}$$ Also, simplify powers inside roots by factoring exponents as multiples of the root index. 3. **Step-by-step solution:** - Express the root: $5\sqrt{32a^5 b^{10}} = 5 \times \sqrt{32} \times \sqrt{a^5} \times \sqrt{b^{10}}$ - Simplify each root: - $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ - $\sqrt{a^5} = \sqrt{a^4 \times a} = a^2 \sqrt{a}$ - $\sqrt{b^{10}} = b^5$ (since $\sqrt{b^{10}} = b^{10/2} = b^5$) - Combine all: $$5 \times 4\sqrt{2} \times a^2 \sqrt{a} \times b^5 = 20 a^2 b^5 \sqrt{2a}$$ 4. **Final answer:** $$5\sqrt{32a^5 b^{10}} = 20 a^2 b^5 \sqrt{2a}$$