1. Stating the problem: Simplify the expression $$\frac{7+\sqrt{3}}{7-\sqrt{3}} + \frac{7-\sqrt{3}}{7+\sqrt{3}}$$ using the least common multiple (LCM) method. 2. Identify the denominators: The denominators are $$7-\sqrt{3}$$ and $$7+\sqrt{3}$$. 3. Find the LCM of the denominators: Since these are conjugates, their LCM is their product: $$\text{LCM} = (7-\sqrt{3})(7+\sqrt{3}) = 7^2 - (\sqrt{3})^2 = 49 - 3 = 46$$ 4. Rewrite each fraction with the common denominator 46: $$\frac{7+\sqrt{3}}{7-\sqrt{3}} = \frac{(7+\sqrt{3})(7+\sqrt{3})}{46} = \frac{(7+\sqrt{3})^2}{46}$$ $$\frac{7-\sqrt{3}}{7+\sqrt{3}} = \frac{(7-\sqrt{3})(7-\sqrt{3})}{46} = \frac{(7-\sqrt{3})^2}{46}$$ 5. Expand the numerators: $$(7+\sqrt{3})^2 = 7^2 + 2 \cdot 7 \cdot \sqrt{3} + (\sqrt{3})^2 = 49 + 14\sqrt{3} + 3 = 52 + 14\sqrt{3}$$ $$(7-\sqrt{3})^2 = 7^2 - 2 \cdot 7 \cdot \sqrt{3} + (\sqrt{3})^2 = 49 - 14\sqrt{3} + 3 = 52 - 14\sqrt{3}$$ 6. Add the two fractions: $$\frac{52 + 14\sqrt{3}}{46} + \frac{52 - 14\sqrt{3}}{46} = \frac{(52 + 14\sqrt{3}) + (52 - 14\sqrt{3})}{46} = \frac{52 + 14\sqrt{3} + 52 - 14\sqrt{3}}{46} = \frac{104}{46}$$ 7. Simplify the fraction: $$\frac{104}{46} = \frac{52}{23}$$ Final answer: $$\boxed{\frac{52}{23}}$$