The expression in the image is:

$\frac{1}{7 + \sqrt{3}} + \frac{1}{7 - \sqrt{3}}$

To simplify this, let's go through the steps:

Step 1: Find a common denominator

The common denominator for the two fractions is $(7 + \sqrt{3})(7 - \sqrt{3})$.

Step 2: Expand the denominator

We use the difference of squares formula: $(a + b)(a - b) = a^2 - b^2$.

$(7 + \sqrt{3})(7 - \sqrt{3}) = 7^2 - (\sqrt{3})^2 = 49 - 3 = 46$

So, the common denominator is $46$.

Step 3: Rewrite the expression with the common denominator

Now, rewrite each fraction as follows:

$\frac{1}{7 + \sqrt{3}} + \frac{1}{7 - \sqrt{3}} = \frac{7 - \sqrt{3}}{46} + \frac{7 + \sqrt{3}}{46}$

Step 4: Combine the fractions

Combine the numerators over the common denominator:

$\frac{7 - \sqrt{3} + 7 + \sqrt{3}}{46} = \frac{14}{46} = \frac{7}{23}$

Final Answer

$\frac{1}{7 + \sqrt{3}} + \frac{1}{7 - \sqrt{3}} = \frac{7}{23}$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. How would you simplify a similar expression like $\frac{1}{5 + \sqrt{2}} + \frac{1}{5 - \sqrt{2}}$?
2. What is the importance of rationalizing the denominator in fractions with square roots?
3. How would you expand and simplify an expression like $(a + b)(a - b)$?
4. What would happen if the denominator had a complex number instead of a square root?
5. Can you find the result of $\frac{1}{a + \sqrt{b}} + \frac{1}{a - \sqrt{b}}$ for any values of $a$ and $b$?

Tip: When working with expressions with square roots, using the difference of squares formula helps in simplifying by eliminating the root in the denominator.