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

The bracket equals $$1 + \frac{1}{4}\left(\frac{a}{b} + \frac{b}{a} - 2\right) = \frac{4ab + a^2 + b^2 - 2ab}{4ab} = \frac{(a+b)^2}{4ab}$$. Its square root is $$\frac{a+b}{2\sqrt{ab}}$$, so the whole expression is $$\frac{2\sqrt{ab}}{a+b} \times \frac{a+b}{2\sqrt{ab}} = 1$$ whatever the values of $$a$$ and $$b$$.