Combine the following expressions. \( \frac{1}{3} \sqrt{45}-\frac{1}{2} \sqrt{12}+\sqrt{20}+\frac{2}{3} \sqrt{27} \) \( 4 \sqrt{5}+\sqrt{3} \) \( 3 \sqrt{5}+\sqrt{3} \) \( 3 \sqrt{3}+\sqrt{5} \)

Let's simplify and combine the expressions step by step! First, we simplify the square roots: - \( \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \) - \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \) - \( \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \) - \( \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \) Now, substituting these back into the original expression: \[ \frac{1}{3}(3\sqrt{5}) - \frac{1}{2}(2\sqrt{3}) + 2\sqrt{5} + \frac{2}{3}(3\sqrt{3}) \] This simplifies to: \[ \sqrt{5} - \sqrt{3} + 2\sqrt{5} + 2\sqrt{3} \] Now, combining like terms: For \( \sqrt{5} \): \[ \sqrt{5} + 2\sqrt{5} = 3\sqrt{5} \] For \( \sqrt{3} \): \[ -\sqrt{3} + 2\sqrt{3} = \sqrt{3} \] Putting it all together, we get: \[ 3\sqrt{5} + \sqrt{3} \] Now let's add the other expressions: Adding \( 4\sqrt{5} + \sqrt{3} \): \[ (3\sqrt{5} + \sqrt{3}) + (4\sqrt{5} + \sqrt{3}) = (3\sqrt{5} + 4\sqrt{5}) + (\sqrt{3} + \sqrt{3}) = 7\sqrt{5} + 2\sqrt{3} \] Adding \( 3\sqrt{5} + \sqrt{3} \): \[ (7\sqrt{5} + 2\sqrt{3}) + (3\sqrt{5} + \sqrt{3}) = (7\sqrt{5} + 3\sqrt{5}) + (2\sqrt{3} + \sqrt{3}) = 10\sqrt{5} + 3\sqrt{3} \] Finally, adding \( 3\sqrt{3} + \sqrt{5} \): \[ (10\sqrt{5} + 3\sqrt{3}) + (3\sqrt{3} + \sqrt{5}) = (10\sqrt{5} + \sqrt{5}) + (3\sqrt{3} + 3\sqrt{3}) = 11\sqrt{5} + 6\sqrt{3} \] Thus, the combined expression is: \[ \boxed{11\sqrt{5} + 6\sqrt{3}} \]