(b) \( \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}-\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} \)

To simplify the expression \( \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}-\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} \), we can use the identity \( a^2 - b^2 = (a-b)(a+b) \). Let \( a = \sqrt{x} + \frac{1}{\sqrt{x}} \) and \( b = \sqrt{x} - \frac{1}{\sqrt{x}} \). Now, calculating \( a-b \) and \( a+b \): 1. \( a - b = \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) - \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right) = \frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x}} = \frac{2}{\sqrt{x}} \) 2. \( a + b = \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) + \left(\sqrt{x} - \frac{1}{\sqrt{x}}\right) = \sqrt{x} + \sqrt{x} = 2\sqrt{x} \) Now applying the identity: \[ \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{2}-\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^{2} = (a-b)(a+b) = \left(\frac{2}{\sqrt{x}}\right) \left(2\sqrt{x}\right) \] Now simplifying this: \[ \frac{2}{\sqrt{x}} \cdot 2\sqrt{x} = 4 \] Thus, the simplified result of the expression is: \[ \boxed{4} \]