1. Simplify the following, expanding brackets where necessary. (i) \( 2 \sqrt{50}-3 \sqrt{27}+4 \sqrt{108} \)

First, let's simplify the square roots: - \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \) - \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \) - \( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \) Now, substituting these back into the equation gives: \( 2\sqrt{50} = 2(5\sqrt{2}) = 10\sqrt{2} \) \( -3\sqrt{27} = -3(3\sqrt{3}) = -9\sqrt{3} \) \( 4\sqrt{108} = 4(6\sqrt{3}) = 24\sqrt{3} \) Now combine these results: \[ 10\sqrt{2} - 9\sqrt{3} + 24\sqrt{3} = 10\sqrt{2} + (24 - 9)\sqrt{3} = 10\sqrt{2} + 15\sqrt{3} \] So, the simplified expression is: \[ 10\sqrt{2} + 15\sqrt{3} \]