Find the product (in simplified form) \( (x \sqrt{y}+z)(x \sqrt{y}-z) \) \( x^{2} y+2 z x \sqrt{y}-z^{2} \) \( x^{2} y^{2}-z^{2} \) \( x^{2} y-z^{2} \) \( x^{2} y-2^{2} \)

To find the product of the expression \( (x \sqrt{y}+z)(x \sqrt{y}-z) \), we can use the difference of squares formula, which states that \( (a + b)(a - b) = a^2 - b^2 \). Here, we have: - \( a = x \sqrt{y} \) - \( b = z \) Applying the formula: \[ (x \sqrt{y} + z)(x \sqrt{y} - z) = (x \sqrt{y})^2 - z^2 \] Now, calculate \( (x \sqrt{y})^2 \): \[ (x \sqrt{y})^2 = x^2 y \] Putting it all together gives us: \[ x^2 y - z^2 \] So, the product in simplified form is: \[ \boxed{x^2 y - z^2} \]