Which of the following radical expressions simplifies to the answer below ? 2x * sqrt(15y) (1 point ) sqrt(30x ^ 2 * y); 5sqrt(6x * y ^ 2); sqrt(60x ^ 2 * y); 3sqrt(50x ^ 2 * y)

Answer 1

To determine which radical expression simplifies to \( 2x \sqrt{15y} \), we'll simplify each option:

1. Option 1: \( \sqrt{30x^2y} \)

   \[ \sqrt{30x^2y} = \sqrt{30} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{30} \cdot x \cdot \sqrt{y} = x \sqrt{30y} \]

   This does not simplify to \( 2x \sqrt{15y} \).

2. Option 2: \( 5\sqrt{6xy^2} \)

   \[ 5\sqrt{6xy^2} = 5\sqrt{6} \cdot \sqrt{x} \cdot \sqrt{y^2} = 5\sqrt{6} \cdot \sqrt{x} \cdot y \ = 5y\sqrt{6x} \]

   This does not simplify to \( 2x \sqrt{15y} \).

3. Option 3: \( \sqrt{60x^2y} \)

   \[ \sqrt{60x^2y} = \sqrt{60} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{60} \cdot x \cdot \sqrt{y} \] Now we can simplify \( \sqrt{60} \): \[ \sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \cdot \sqrt{15} = 2\sqrt{15} \] Thus, \[ \sqrt{60x^2y} = 2\sqrt{15} \cdot x \cdot \sqrt{y} = 2x \sqrt{15y} \]

   This simplifies to \( 2x \sqrt{15y} \).

4. Option 4: \( 3\sqrt{50x^2y} \)

   \[ 3\sqrt{50x^2y} = 3\sqrt{50} \cdot \sqrt{x^2} \cdot \sqrt{y} = 3\sqrt{50} \cdot x \cdot \sqrt{y} \] Simplifying \( \sqrt{50} \): \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \] Thus, \[ 3\sqrt{50x^2y} = 3 \cdot 5\sqrt{2} \cdot x \cdot \sqrt{y} = 15x\sqrt{2y} \]

   This does not simplify to \( 2x \sqrt{15y} \).

Since only Option 3: \( \sqrt{60x^2y} \) simplifies to \( 2x \sqrt{15y} \), that is the correct answer.