Question

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:
\[
\boxed{8x^2y^4\sqrt{2y}}
\]