2. If \( \sqrt{u}+\sqrt{v}-\sqrt{u}=0 \). find the value of \( (u+v-w+2 \sqrt{u v}) \) 2. Find the product of \( \left(x-\frac{1}{x}\right)\left(x+\frac{1}{x}\right)\left(x^{2}+\frac{1}{x^{2}}\right)\left(x^{4}+\frac{1}{x^{4}}\right) \) using a suitable identity.

Let's solve each problem step by step. ### Problem 1: We start with the equation: \[ \sqrt{u} + \sqrt{v} - \sqrt{u} = 0 \] This simplifies to: \[ \sqrt{v} = 0 \] Thus, we have: \[ v = 0 \] Now, we need to find the value of \( (u + v - w + 2\sqrt{uv}) \). Since \( v = 0 \), we can substitute this into the expression: \[ u + v - w + 2\sqrt{uv} = u + 0 - w + 2\sqrt{u \cdot 0} \] This simplifies to: \[ u - w + 0 = u - w \] Therefore, the value of \( (u + v - w + 2\sqrt{uv}) \) is: \[ \boxed{u - w} \] ### Problem 2: We need to find the product: \[ \left(x - \frac{1}{x}\right)\left(x + \frac{1}{x}\right)\left(x^2 + \frac{1}{x^2}\right)\left(x^4 + \frac{1}{x^4}\right) \] First, we can simplify the first two factors: \[ \left(x - \frac{1}{x}\right)\left(x + \frac{1}{x}\right) = x^2 - \left(\frac{1}{x}\right)^2 = x^2 - \frac{1}{x^2} = \frac{x^4 - 1}{x^2} \] Next, we can express \( x^2 + \frac{1}{x^2} \) in terms of \( x^2 - \frac{1}{x^2} \): \[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \] Let \( a = x - \frac{1}{x} \). Then: \[ x^2 + \frac{1}{x^2} = a^2 + 2 \] Now, we can also express \( x^4 + \frac{1}{x^4} \): \[ x^4 + \frac{1}{x^4} = \left(x^2 - \frac{1}{x^2}\right)^2 + 2 = \left(a^2 - 2\right)^2 + 2 \] Now, we can combine everything: \[ \left(x - \frac{1}{x}\right)\left(x + \frac{1}{x}\right) = \frac{x^4 - 1}{x^2} \] Thus, the product becomes: \[ \frac{x^4 - 1}{x^2} \cdot \left(x^2 + \frac{1}{x^2}\right) \cdot \left(x^4 + \frac{1}{x^4}\right) \] Now substituting \( x^2 + \frac{1}{x^2} \) and \( x^4 + \frac{1}{x^4} \) back into the product, we can simplify further, but it is easier to compute directly: \[ \left(x^2 - \frac{1}{x^2}\right)\left(x^2 + \frac{1}{x^2}\right) = x^4 - \frac{1}{x^4} \] Thus, the entire product simplifies to: \[ \left(x^4 - 1\right)\left(x^4 + 1\right) = x^8 - 1 \] Therefore, the final result is: \[ \boxed{x^8 - 1} \]