$\begin{array}{l} A = (\frac{{\sqrt a + \sqrt b }}{{\sqrt a - \sqrt b }} - \frac{{\sqrt a - \sqrt b }}{{\sqrt a + \sqrt b }})(a - b)\\ = \frac{{{{(\sqrt a + \sqrt b )}^2} - {{(\sqrt a - \sqrt b )}^2}}}{{(\sqrt a - \sqrt b )(\sqrt a + \sqrt b )}}.(a - b)\\ = \frac{{(a + 2\sqrt a \sqrt b + b) - (a - 2\sqrt a \sqrt b + b)}}{{a - b}}(a - b)\\ = 4\sqrt {ab} \end{array}$
