Rationalize the denominator and simplify. Assume that all variables represent positive real numbers.
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Question

Quizplus · Accepted Answer

To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{ab} - \sqrt{cd}$. This gives us:
$$\frac{(\sqrt{ab} + \sqrt{cd})(\sqrt{ab} - \sqrt{cd})}{(\sqrt{ab} + \sqrt{cd})(\sqrt{ab} - \sqrt{cd})} = \frac{ab - cd}{ab - cd + 2\sqrt{abcd}}$$
Since all variables represent positive real numbers, $ab > cd$. Therefore, $ab - cd$ and $2\sqrt{abcd}$ are both positive, so we can simplify by factoring out the common factor of $ab - cd$:
$$\frac{ab - cd}{ab - cd + 2\sqrt{abcd}} = \frac{1}{1 + \frac{2\sqrt{abcd}}{ab - cd}}$$
Therefore, the answer is $\boxed{	ext{C}}$. To rationalize the denominator, we need to multiply both the numerator and denominator by $\sqrt{5} - \sqrt{3}$. This gives us:
$$\frac{(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})} = \frac{5 - 3}{5 - 3} = \boxed{1}$$
Therefore, the correct answer is C.

Rationalize the Denominator and Simplify