1. **State the problem:** Find the value of $$k^2 + \frac{1}{k^2}$$ where $$k = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$$. 2. **Formula and rules:** We will first simplify $$k$$ by rationalizing the denominator. Recall that $$\frac{a+b}{a-b} = \frac{(a+b)(a+b)}{(a-b)(a+b)} = \frac{(a+b)^2}{a^2 - b^2}$$. 3. **Simplify $$k$$:** $$k = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2} = \frac{(\sqrt{5})^2 + 2\sqrt{5}\sqrt{3} + (\sqrt{3})^2}{5 - 3}$$ 4. **Calculate numerator and denominator:** $$= \frac{5 + 2\sqrt{15} + 3}{2} = \frac{8 + 2\sqrt{15}}{2}$$ 5. **Simplify fraction:** $$= \cancel{\frac{2}{2}}(4 + \sqrt{15}) = 4 + \sqrt{15}$$ 6. **Calculate $$k^2$$:** $$k^2 = (4 + \sqrt{15})^2 = 4^2 + 2 \times 4 \times \sqrt{15} + (\sqrt{15})^2 = 16 + 8\sqrt{15} + 15 = 31 + 8\sqrt{15}$$ 7. **Calculate $$\frac{1}{k}$$:** $$\frac{1}{k} = \frac{1}{4 + \sqrt{15}}$$ Rationalize denominator: $$\frac{1}{4 + \sqrt{15}} \times \frac{4 - \sqrt{15}}{4 - \sqrt{15}} = \frac{4 - \sqrt{15}}{16 - 15} = 4 - \sqrt{15}$$ 8. **Calculate $$\frac{1}{k^2}$$:** $$\left(\frac{1}{k}\right)^2 = (4 - \sqrt{15})^2 = 16 - 8\sqrt{15} + 15 = 31 - 8\sqrt{15}$$ 9. **Calculate $$k^2 + \frac{1}{k^2}$$:** $$= (31 + 8\sqrt{15}) + (31 - 8\sqrt{15}) = 31 + 31 + 8\sqrt{15} - 8\sqrt{15} = 62$$ **Final answer:** $$\boxed{62}$$