Exercise 14. Let $x$ be a rational number such that $\sqrt{x}+\sqrt{x+13}=\sqrt{4 x+17}$. We write $x$ in the form $x=\frac{a}{b}$ where $a$ and $b$ are natural numbers with no common divisors. Determine the value of $a+b$.

---

The translation is provided as requested, maintaining the original formatting and structure.
Solution to exercise 14: We square and simplify: $2 \sqrt{x^{2}+13 x}=2 x+4$. We square again and simplify, which gives $x=4 / 9$. Conversely, we verify that $x$ works, so $a+b=13$.