1. **State the problem:** Simplify the expression $$\sqrt{13} + \frac{3}{\sqrt{13} - 3}$$. 2. **Recall the formula:** To simplify expressions with radicals in the denominator, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$\sqrt{13} - 3$$ is $$\sqrt{13} + 3$$. 3. **Rationalize the denominator:** $$\frac{3}{\sqrt{13} - 3} \times \frac{\sqrt{13} + 3}{\sqrt{13} + 3} = \frac{3(\sqrt{13} + 3)}{(\sqrt{13} - 3)(\sqrt{13} + 3)}$$ 4. **Simplify the denominator using difference of squares:** $$(\sqrt{13})^2 - 3^2 = 13 - 9 = 4$$ 5. **Rewrite the fraction:** $$\frac{3\sqrt{13} + 9}{4}$$ 6. **Combine with the first term:** $$\sqrt{13} + \frac{3\sqrt{13} + 9}{4} = \frac{4\sqrt{13}}{4} + \frac{3\sqrt{13} + 9}{4} = \frac{4\sqrt{13} + 3\sqrt{13} + 9}{4}$$ 7. **Combine like terms:** $$\frac{(4 + 3)\sqrt{13} + 9}{4} = \frac{7\sqrt{13} + 9}{4}$$ **Final answer:** $$\frac{7\sqrt{13} + 9}{4}$$