To simplify the expression \(\frac{5+\sqrt{7}}{3+\sqrt{7}}\), we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is \(3-\sqrt{7}\). The conjugate is obtained by changing the sign between the terms involving the square root.

\(\frac{5+\sqrt{7}}{3+\sqrt{7}} \cdot \frac{3-\sqrt{7}}{3-\sqrt{7}}\)

Now, we can use the distributive property to multiply the numerators and the denominators:

\(\frac{(5+\sqrt{7})(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}\)

Using the distributive property again, we can expand the numerators and denominators:

\(\frac{15-5\sqrt{7}+3\sqrt{7}-7}{9-7}\)

Combine the like terms in the numerator:

\(\frac{8-2\sqrt{7}}{2}\)

Divide both the numerator and the denominator by 2:

\(\frac{4-\sqrt{7}}{1}\)

So, the simplified form of \(\frac{5+\sqrt{7}}{3+\sqrt{7}}\) is \(4-\sqrt{7}\).