To simplify the expression $\frac{5+\sqrt{7}}{3+\sqrt{7}}$, we need to rationalize the denominator. To do this, we can multiply the numerator and denominator by the conjugate of the denominator, which is $3-\sqrt{7}$. This gives us:
$$\frac{5+\sqrt{7}}{3+\sqrt{7}}\cdot\frac{3-\sqrt{7}}{3-\sqrt{7}}=\frac{(5+\sqrt{7})(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}=\frac{15-5\sqrt{7}+3\sqrt{7}-7}{9-7}$$
Simplifying the numerator and denominator gives:
$$\frac{8-2\sqrt{7}}{2}=\boxed{4-\sqrt{7}}$$
Therefore, $\frac{5+\sqrt{7}}{3+\sqrt{7}}$ simplifies to $4-\sqrt{7}$.