[tex]A=\frac{a(\frac{2b\sqrt{a}}{\sqrt{a}+\sqrt{b}})+b(\frac{2a\sqrt{b}}{\sqrt{a}+\sqrt{b}})}{\frac{2ab}{\sqrt{a}(\sqrt{a}+\sqrt{b})}+\frac{2ab}{\sqrt{b}(\sqrt{a}+\sqrt{b})}}=\frac{\frac{2ab(\sqrt{a}+\sqrt{b})}{\sqrt{a}+\sqrt{b}}}{2ab(\frac{\sqrt{a}+\sqrt{b}}{\sqrt{ab}(\sqrt{a}+\sqrt{b})})}=\sqrt{ab}[/tex]