Rationalize the denominator of \( \frac{\sqrt{a}+4}{2 \sqrt{a}-\sqrt{b}} \). Assume that all variables represent positive real numbers. \( \frac{\sqrt{a}+4}{2 \sqrt{a}-\sqrt{b}}=\square \)

To rationalize the denominator of \( \frac{\sqrt{a}+4}{2 \sqrt{a}-\sqrt{b}} \), we multiply the numerator and denominator by the conjugate of the denominator, \( 2 \sqrt{a} + \sqrt{b} \): \[ \frac{(\sqrt{a}+4)(2 \sqrt{a} + \sqrt{b})}{(2 \sqrt{a}-\sqrt{b})(2 \sqrt{a}+\sqrt{b})} \] The denominator simplifies as follows: \[ (2 \sqrt{a})^2 - (\sqrt{b})^2 = 4a - b \] Now, expand the numerator: \[ (\sqrt{a}+4)(2 \sqrt{a}+\sqrt{b}) = 2a + \sqrt{ab} + 8\sqrt{a} + 4\sqrt{b} = 2a + 8\sqrt{a} + \sqrt{ab} + 4\sqrt{b} \] Putting it all together, we get: \[ \frac{2a + 8\sqrt{a} + \sqrt{ab} + 4\sqrt{b}}{4a - b} \] Thus, the rationalized form is: \[ \frac{2a + 8\sqrt{a} + \sqrt{ab} + 4\sqrt{b}}{4a - b} \]