27. \( (2 \sqrt{x}-3 \sqrt{y})(3 \sqrt{x}+5 \sqrt{y})= \)

To simplify the expression \( (2 \sqrt{x}-3 \sqrt{y})(3 \sqrt{x}+5 \sqrt{y}) \), we can use the distributive property (also known as the FOIL method for binomials): 1. **First:** Multiply the first terms: \( 2 \sqrt{x} \cdot 3 \sqrt{x} = 6x \). 2. **Outer:** Multiply the outer terms: \( 2 \sqrt{x} \cdot 5 \sqrt{y} = 10 \sqrt{xy} \). 3. **Inner:** Multiply the inner terms: \( -3 \sqrt{y} \cdot 3 \sqrt{x} = -9 \sqrt{xy} \). 4. **Last:** Multiply the last terms: \( -3 \sqrt{y} \cdot 5 \sqrt{y} = -15y \). Now, combine all the results together: \[ 6x + 10 \sqrt{xy} - 9 \sqrt{xy} - 15y. \] This simplifies to: \[ 6x + (10\sqrt{xy} - 9\sqrt{xy}) - 15y = 6x + \sqrt{xy} - 15y. \] So, the final simplified expression is: \[ 6x + \sqrt{xy} - 15y. \]