sqrt (50)+sqrt (12)-4sqrt (18)+4sqrt (27)

To simplify the expression \(\sqrt{50} + \sqrt{12} - 4\sqrt{18} + 4\sqrt{27}\), we first express each square root in terms of its prime factors:<br /><br />\[<br />\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}<br />\]<br /><br />\[<br />\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}<br />\]<br /><br />\[<br />4\sqrt{18} = 4 \cdot \sqrt{9 \times 2} = 4 \cdot \sqrt{9} \cdot \sqrt{2} = 4 \cdot 3\sqrt{2} = 12\sqrt{2}<br />\]<br /><br />\[<br />4\sqrt{27} = 4 \cdot \sqrt{9 \times 3} = 4 \cdot \sqrt{9} \cdot \sqrt{3} = 4 \cdot 3\sqrt{3} = 12\sqrt{3}<br />\]<br /><br />Now, substitute these simplified forms back into the original expression:<br /><br />\[<br />5\sqrt{2} + 2\sqrt{3} - 12\sqrt{2} + 12\sqrt{3}<br />\]<br /><br />Combine like terms:<br /><br />\[<br />(5\sqrt{2} - 12\sqrt{2}) + (2\sqrt{3} + 12\sqrt{3})<br />\]<br /><br />\[<br />= -7\sqrt{2} + 14\sqrt{3}<br />\]<br /><br />Thus, the simplified form of the expression is:<br /><br />\[<br />\boxed{-7\sqrt{2} + 14\sqrt{3}}<br />\]