If $$a = \sqrt{2024}$$, $$b = \sqrt{2025}$$, the value of $$2(ab)^{1/2}(a + b)^{-1}\left\{1 + \frac{1}{4}\left(\sqrt{\frac{a}{b}} - \sqrt{\frac{b}{a}}\right)^2\right\}^{1/2}$$ is

$$1 + \dfrac{1}{4}\left(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{b}{a}}\right)^2$$ can be expanded using $$(a+b)^2$$ formula to get

 $$1 + \dfrac{1}{4}\left(\dfrac{a}{b} + \dfrac{b}{a} - 2\right) = \dfrac{4ab + a^2 + b^2 - 2ab}{4ab} = \dfrac{(a+b)^2}{4ab}$$. 

As $$a,b >0$$, its square root is 

$$\dfrac{a+b}{2\sqrt{ab}}$$

Now, the expression reduces to 

$$\dfrac{2\sqrt{ab}}{a+b} \times \dfrac{a+b}{2\sqrt{ab}} = 1$$