1. The problem is to rationalize the denominator of the fraction $$\frac{7}{4 + \sqrt{7}}$$. 2. To rationalize the denominator, we multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$4 + \sqrt{7}$$ is $$4 - \sqrt{7}$$. 3. Multiply numerator and denominator: $$\frac{7}{4 + \sqrt{7}} \times \frac{4 - \sqrt{7}}{4 - \sqrt{7}} = \frac{7(4 - \sqrt{7})}{(4 + \sqrt{7})(4 - \sqrt{7})}$$ 4. Simplify the denominator using the difference of squares formula: $$(4 + \sqrt{7})(4 - \sqrt{7}) = 4^2 - (\sqrt{7})^2 = 16 - 7 = 9$$ 5. Expand the numerator: $$7(4 - \sqrt{7}) = 28 - 7\sqrt{7}$$ 6. Write the simplified expression: $$\frac{28 - 7\sqrt{7}}{9}$$ 7. This is the rationalized form of the original expression. Final answer: $$\frac{28 - 7\sqrt{7}}{9}$$