1. The problem is to simplify the expression $$\sqrt[8]{16a^8b^4}$$ and verify if it equals $$\sqrt{2a^2b}$$. 2. Recall the rule for radicals: $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$. 3. Rewrite each part inside the eighth root using exponents: $$16 = 2^4$$ $$a^8$$ stays as is $$b^4$$ stays as is 4. So, $$\sqrt[8]{16a^8b^4} = \sqrt[8]{2^4 \cdot a^8 \cdot b^4} = 2^{\frac{4}{8}} \cdot a^{\frac{8}{8}} \cdot b^{\frac{4}{8}} = 2^{\frac{1}{2}} \cdot a^{1} \cdot b^{\frac{1}{2}}$$ 5. Simplify the exponents: $$2^{\frac{1}{2}} = \sqrt{2}$$ $$a^{1} = a$$ $$b^{\frac{1}{2}} = \sqrt{b}$$ 6. Combine the terms: $$\sqrt{2} \cdot a \cdot \sqrt{b} = a \sqrt{2b}$$ 7. The original expression simplifies to $$a \sqrt{2b}$$, which is not equal to $$\sqrt{2a^2b}$$. 8. Note that $$\sqrt{2a^2b} = \sqrt{2} \cdot \sqrt{a^2} \cdot \sqrt{b} = \sqrt{2} \cdot a \cdot \sqrt{b} = a \sqrt{2b}$$. 9. Therefore, both expressions are equal: $$\sqrt[8]{16a^8b^4} = \sqrt{2a^2b} = a \sqrt{2b}$$.