Simplify the expression. \( \sqrt[4]{768 x^{8} y^{5}} \) A. \( 4 x^{2} y \sqrt[4]{3 y} \) B. \( 4 x^{2} y^{2} \sqrt[4]{3 x^{2} y} \) C. \( 2 x y^{2} \sqrt[4]{2 x} \) D. \( 4 x^{3} y \sqrt[4]{3 x^{2}} \)

To simplify the expression \( \sqrt[4]{768 x^{8} y^{5}} \), we first break down \( 768 \) into its prime factors. We can factor \( 768 \) as follows: \[ 768 = 256 \times 3 = 2^8 \times 3 \] Now, we rewrite the expression: \[ \sqrt[4]{768 x^{8} y^{5}} = \sqrt[4]{2^8 \times 3 \times x^{8} \times y^{5}} \] Next, we can separate the radical: \[ \sqrt[4]{2^8} \times \sqrt[4]{3} \times \sqrt[4]{x^8} \times \sqrt[4]{y^5} \] By simplifying each part: 1. \( \sqrt[4]{2^8} = 2^2 = 4 \) 2. \( \sqrt[4]{x^8} = x^2 \) 3. For \( y^{5} \), we can write \( y^5 \) as \( y^4 \cdot y = (y^4)^{1/4} \cdot y^{1/4}= y \sqrt[4]{y} \) Putting it all together, we have: \[ 4 x^2 y \sqrt[4]{3 y} \] So the simplified expression is: A. \( 4 x^{2} y \sqrt[4]{3 y} \) Now, if we double-check our calculation: - We broke down the components correctly, - Combined them logically, - Final checks align with our original factors. Therefore, the answer is A. \( 4 x^{2} y \sqrt[4]{3 y} \).