To rationalize the denominator of the given expression:

$\frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{7} - \sqrt{5}}$

We will multiply both terms by their respective conjugates.

Step 1: Rationalize the first term

$\frac{1}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}} = \frac{\sqrt{7} - \sqrt{5}}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{\sqrt{7} - \sqrt{5}}{7 - 5} = \frac{\sqrt{7} - \sqrt{5}}{2}$

Step 2: Rationalize the second term

$\frac{1}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{\sqrt{7} + \sqrt{5}}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{\sqrt{7} + \sqrt{5}}{7 - 5} = \frac{\sqrt{7} + \sqrt{5}}{2}$

Step 3: Combine the two expressions

Now, adding the two rationalized expressions:

$\frac{\sqrt{7} - \sqrt{5}}{2} + \frac{\sqrt{7} + \sqrt{5}}{2} = \frac{(\sqrt{7} - \sqrt{5}) + (\sqrt{7} + \sqrt{5})}{2} = \frac{2\sqrt{7}}{2} = \sqrt{7}$

Final Answer

The rationalized and simplified result of the given expression is:

$\sqrt{7}$

Would you like more details or have any further questions?

Here are some related questions:

1. What are the steps to rationalize a denominator in general?
2. How does multiplying by the conjugate help in rationalizing the denominator?
3. Can this method be used for any expression with a square root in the denominator?
4. What would happen if the denominator had a more complex expression, like $\sqrt{a} + \sqrt{b}$?
5. How does rationalizing the denominator affect the value of an expression?

Tip: When rationalizing a denominator, always check if multiplying by the conjugate simplifies the expression completely.