1. **State the problem:** Simplify the expression $$\sqrt{2 + \frac{5}{\sqrt{3}}}$$. 2. **Rewrite the expression:** The expression inside the square root is $$2 + \frac{5}{\sqrt{3}}$$. 3. **Rationalize the denominator:** To simplify $$\frac{5}{\sqrt{3}}$$, multiply numerator and denominator by $$\sqrt{3}$$: $$\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{\cancel{\sqrt{3}}\sqrt{3}} = \frac{5\sqrt{3}}{3}$$. 4. **Rewrite the expression inside the root:** $$2 + \frac{5\sqrt{3}}{3} = \frac{6}{3} + \frac{5\sqrt{3}}{3} = \frac{6 + 5\sqrt{3}}{3}$$. 5. **Simplify the entire expression:** $$\sqrt{\frac{6 + 5\sqrt{3}}{3}} = \frac{\sqrt{6 + 5\sqrt{3}}}{\sqrt{3}}$$. 6. **Check if the numerator can be simplified:** Try to express $$6 + 5\sqrt{3}$$ as $$(a + b\sqrt{3})^2 = a^2 + 2ab\sqrt{3} + 3b^2$$. Set up equations: $$a^2 + 3b^2 = 6$$ $$2ab = 5$$ From $$2ab=5$$, $$ab=\frac{5}{2}$$. Try $$a=\frac{5}{2b}$$, substitute into first equation: $$\left(\frac{5}{2b}\right)^2 + 3b^2 = 6$$ $$\frac{25}{4b^2} + 3b^2 = 6$$ Multiply both sides by $$4b^2$$: $$25 + 12b^4 = 24b^2$$ Rearranged: $$12b^4 - 24b^2 + 25 = 0$$ This quadratic in $$b^2$$ has no real roots (discriminant negative), so no simplification. 7. **Final simplified form:** $$\boxed{\frac{\sqrt{6 + 5\sqrt{3}}}{\sqrt{3}}}$$. This is the simplest exact form of the expression.