1. The problem is to simplify the fifth root expression $$\sqrt[5]{3^{15} a^{10} b^{20}}$$ and identify which of the given options matches the simplified form. 2. Recall the property of radicals: $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$, so apply this to each factor inside the root: $$\sqrt[5]{3^{15}} = 3^{\frac{15}{5}} = 3^3 = 27$$ $$\sqrt[5]{a^{10}} = a^{\frac{10}{5}} = a^2$$ $$\sqrt[5]{b^{20}} = b^{\frac{20}{5}} = b^4$$ 3. Multiply these simplified terms together: $$27 \times a^2 \times b^4 = 27 a^2 b^4$$ 4. Compare the result with the given options: - 9 a^5 b^{10} - 27 a^2 b^4 - 3 a^4 b^8 The simplified expression matches the second option. Final answer: $$27 a^2 b^4$$