1. **Problem Statement:** Prove that $2 + 3\sqrt{5}$ is an irrational number, given that $\sqrt{3}$ is irrational. 2. **Recall Definitions:** - A number is **rational** if it can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q \neq 0$. - A number is **irrational** if it cannot be expressed as such a fraction. 3. **Given:** $\sqrt{3}$ is irrational. This is a known fact and helps us understand the nature of square roots of non-perfect squares. 4. **Goal:** Show $2 + 3\sqrt{5}$ is irrational. 5. **Proof by Contradiction:** Assume $2 + 3\sqrt{5}$ is rational. 6. Then $2 + 3\sqrt{5} = r$ where $r$ is rational. 7. Rearranging, $3\sqrt{5} = r - 2$. 8. Since $r$ and $2$ are rational, $r - 2$ is rational. 9. Therefore, $\sqrt{5} = \frac{r - 2}{3}$ is rational. 10. But $\sqrt{5}$ is known to be irrational (like $\sqrt{3}$), so this is a contradiction. 11. Hence, $2 + 3\sqrt{5}$ must be irrational. **Final answer:** $2 + 3\sqrt{5}$ is irrational.