Simplify the radical sqrt(128x ^ 4 * y ^ 9) (1 point) 8x ^ 2 * y ^ 4 * sqrt(2y); 8x ^ 3 * y ^ 6 * sqrt(2x * y ^ 3); 14x ^ 3 * y ^ 6 * sqrt(2x * y ^ 3); 14x ^ 2 * y ^ 4 * sqrt(6y)

Answer 1

To simplify the expression \(\sqrt{128x^4y^9}\), we can break it down as follows:

1. Factor the radicand: \[ \sqrt{128} \times \sqrt{x^4} \times \sqrt{y^9} \]

2. Simplify each part:

   • For \(\sqrt{128}\):

     • \(128 = 64 \times 2 = 8^2 \times 2\)
     • Thus, \(\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}\)

   • For \(\sqrt{x^4}\):

     • \(\sqrt{x^4} = x^{4/2} = x^2\)

   • For \(\sqrt{y^9}\):

     • \(y^9 = y^{8+1} = y^8 \times y = (y^4)^2 \times y\)
     • Thus, \(\sqrt{y^9} = \sqrt{y^8 \times y} = y^4 \times \sqrt{y}\)

3. Combine all parts: \[ \sqrt{128x^4y^9} = 8\sqrt{2} \times x^2 \times y^4 \times \sqrt{y} = 8x^2y^4\sqrt{2y} \]

Thus, the simplified radical is: \[ 8x^2y^4\sqrt{2y} \]

From the options provided, the correct answer is: \[ \boxed{8x^2y^4\sqrt{2y}} \]